Question:
Find the surface area of the pyramid. The perpendicular height is 8mm.

Round your answer to two decimal places.

Answer: 288.6 \mathrm{m}^2

1. Given that the base is a square with a side length of 10mm, according to the 'Regular Polygon Area Formula', the area of the square is equal to the square of its side length, i.e., 10mm × 10mm = 100 square millimeters.

2. The four lateral faces are identical isosceles triangles, and the vertical height from the apex is known to be 8mm.

3. In the lateral face, a right triangle can be formed from the midpoint of the base to the endpoint, which can be used to apply the Pythagorean theorem to aid in the calculation. Half of the base is 5mm. Using the 'Pythagorean Theorem' a² + b² = c², here a is the height 8mm, b is half of the base 5mm, and c is the unknown hypotenuse length.

4. Solving the equation 8² + 5² = c², which equals 64 + 25 = c², c² = 89, we find that c, the hypotenuse length, is √89, approximately 9.43mm.

5. According to the 'Triangle Area Formula', the area of each triangle is equal to 1/2 × base length (10mm) × height (calculated hypotenuse 9.43mm), approximately 47.15 square millimeters.

6. The total surface area is equal to the base area plus the area of the four lateral faces, 100 square millimeters + 4×47.15 square millimeters = 100 square millimeters + 188.60 square millimeters = 288.60 square millimeters.

7. Following the above reasoning steps, the final answer is that the surface area is 288.60 square millimeters.

Definition of Regular Polygon

A polygon is a regular polygon if and only if all its sides are of equal length and all its interior angles are equal.

<note>The base is a square</note>, for this square, <note>each of its sides is 10 millimeters long</note>, and <note>each interior angle is 90 degrees</note>. Therefore, <note>the base is a regular polygon</note>.

Definition of Isosceles Triangle

<note>Four sides are all isosceles triangles</note>, each isosceles triangle's <note>two equal sides are respectively the hypotenuse (approximately 9.43 millimeters) and the other hypotenuse (approximately 9.43 millimeters)</note>.

Area Formula for Square

The area of a square is equal to the square of its side length, that is, Area = side length × side length.

In this problem, the <note>area of the square base</note> is <note>10 millimeters × 10 millimeters</note> = <note>100 square millimeters</note>.

Pythagorean Theorem

In this problem, the side of the isosceles triangle has <note>half of the base as 5 mm</note>, <note>the height is 8 mm</note>, so we use the Pythagorean Theorem to calculate <note>the length of the hypotenuse c</note>: a² + b² = c², where a is <note>8 mm</note>, b is <note>5 mm</note>, and c is <note>the length of the hypotenuse</note>. According to the Pythagorean Theorem, <note>8² + 5² = c²</note>, i.e., <note>c² = 89</note>, resulting in <note>the length of the hypotenuse being √89 ≈ 9.43 mm</note>.

Area Formula of a Triangle

The area of any triangle is equal to its base multiplied by its height, divided by 2.

Original: <note>In an isosceles triangle, the base is 10 millimeters</note>, <note>the vertical height is 8 millimeters</note>. According to the area formula of a triangle, the area of the triangle is equal to <note>base 10 millimeters multiplied by height 9.43 millimeters then divided by 2</note>, i.e., area = (<note>10 millimeters * 9.43 millimeters</note>) / 2.


Question: Calculate the radius of the sphere shown in figure with the volume of the cylinder 13.75\pi cm3.

Answer: 3

1. Given the volume of the cylinder V_c = 13.75\pi\text{ cm}^3, and V_c = \pi r^2 h, where r is the radius of the base of the cylinder, and h is the height of the cylinder.

2. The known height of the cylinder is h = 5\text{ cm}. Substituting into the volume formula, we have \pi r^2 \times 5 = 13.75\pi.

3. Removing \pi from both sides of the volume formula, the equation becomes 5r^2 = 13.75.

4. Solving the equation: r^2 = \frac{13.75}{5} = 2.75, thus r = \sqrt{2.75} = \frac{\sqrt{11}}{2}.

5. According to the problem conditions, the cylinder is considered to be embedded in a sphere. From the figure, based on the Pythagorean theorem, we can deduce that the square of the diameter of the sphere equals the sum of the square of the diameter of the base circle of the cylinder and the square of the height. Therefore, diameter = \sqrt{({2r})^2 + h^2}.

6. Substituting the values of r and h, we calculate the diameter = \sqrt{(2\times\frac{\sqrt{11}}{2})^2 + 5^2} = \sqrt{11 + 25} = 6.

7. Therefore, the radius of the sphere R = \frac{6}{2} = 3.

8. Finally, from the above calculations, the radius of the sphere is 3.

Volume Formula of Cylinder

The volume of a cylinder is equal to the area of its base multiplied by its height. \( V = \pi r^2 h \), where \( r \) is the radius of the base of the cylinder, and \( h \) is the height of the cylinder.

Original: <note>Volume of Cylinder</note> V_c = 13.75π cm^3, <note>Height of Cylinder</note> h = 5 cm. Substituting these values into the formula, we obtained <note>Base Radius</note> r = √2.75 = \frac{√11}{2} cm, and proceeded with subsequent calculations.

Radius of a Sphere

The radius R of a sphere is the distance from the center of the sphere to any point on its surface.

<note>We use the diameter 2r of the cylinder's base and the height h</note>, and apply the Pythagorean theorem to find <note>the diameter of the embedded sphere d = √[(2r)^2 + h^2]</note> = √[11 + 25] = 6 cm, thus we can obtain <note>the radius of the sphere R = rac{d}{2} = 3 cm</note>.

Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs.

Original: <note>The height of the cylinder h = 5 cm</note>, <note>another leg is the diameter of the cylinder's base 2r = 2 × rac{√11}{2} = √11 cm</note>, <note>the hypotenuse is the diameter of the sphere d</note>. Therefore, using the Pythagorean Theorem, we can calculate <note>the diameter of the sphere d = √[(2r)^2 + h^2] = 6 cm</note>.


Question: As shown in the figure, the height of the prism is 9cm. Find the volume of the prism shown.

Answer: 270 \mathrm{cm}^3

1. Given: The height of the prism is 9 cm, and the area of the base A = 30 cm².

2. The volume formula for the prism is: Volume = base area × height.

3. Substitute the given conditions into the volume formula: Volume = 30 cm² × 9 cm.

4. Calculate: Volume = 270 cm³.

5. After the above calculations, the final answer is 270 cm³.

Definition of Prism

A prism is a polyhedron with two parallel and congruent polygonal bases, and all other faces are parallelograms.

<note>In the figure of this problem, the two identical bases of the prism are the two faces of the pentagon</note>, <note>and the lateral faces are the rectangular faces connecting the edges of the two bases</note>.

Volume Formula of Prism

The volume of a prism is equal to the base area multiplied by the height.

<note>Substitute the known base area A = 30 square centimeters and height h = 9 centimeters into the volume formula</note>, volume = <note>base area × height = 30 square centimeters × 9 centimeters</note> = 270 cubic centimeters.


Question:
Find the surface area of the cone shown. The radius is 3cm.

Round your answer to two decimal places.

Answer: 122.52 \mathrm{cm}^2

1. Given that the base radius r of the cone is 3 cm, and the slant height l is 10 cm.

2. To calculate the surface area of the cone, we first need to calculate the area of the base circle. The formula for the area of the base circle is A_circle = πr^2.

3. Substituting the given conditions, r=3 cm, we can obtain the area of the base circle A_circle = π × (3)^2 = 9π cm².

4. Next, we calculate the lateral area. The formula for the lateral area is A_side = πrl, which equals the circumference of the base circle πr multiplied by the slant height l.

5. Substituting the given conditions, r=3 cm and l=10 cm, we can obtain the lateral area A_side = π × 3 × 10 = 30π cm².

6. Adding the base area and the lateral area together, we get the total surface area S_total = A_circle + A_side = 9π + 30π = 39π cm².

7. Calculating the approximate value of 39π, we get S_total = 39 × 3.14159 ≈ 122.522113 cm².

8. Rounding the final calculated value to two decimal places, 122.522113 ≈ 122.52 cm².

9. Through the above reasoning, the final answer is: The surface area of the cone is 122.52 cm².

Definition of Cone

A cone in three-dimensional space is a surface generated by rotating a line segment (generatrix) around a fixed straight line (axis) belonging to a plane of a circle, which is not collinear with the endpoint (vertex). The surface consists of the conical surface and the base (circle).

In the figure of this problem, <note>the plane base of the cone is the base circle</note>, its <note>radius is r=3 cm</note>, <note>the vertex is the top of the cone</note>, and the line segment between the base circle and the vertex represents <note>the slant height l=10 cm</note>.

Area Formula of a Circle

The area of a circle is given by the formula A = πr², where A represents the area, π (pi) is the mathematical constant approximately equal to 3.14159, and r is the radius of the circle.

<note>The radius of the base circle of the cone is 3 cm</note>, according to the area formula of a circle, <note>the area A of the circle is equal to pi multiplied by the square of the radius 3</note>, that is, <note>A_circle = π × 3² = 9π cm²</note>.

Lateral Surface Area of a Cone

The lateral surface area \(A\) of a cone is calculated using the formula \(A = \pi r l\), where \(r\) is the radius of the base circle and \(l\) is the slant height of the cone.

The original: <note>The radius of the base circle r=3 cm</note>, <note>the slant height of the cone l=10 cm</note>, therefore <note>the lateral surface area A_side = π × 3 × 10 = 30π cm²</note>.

Formula for the Surface Area of a Cone

The total surface area of a cone is equal to the sum of the base area and the lateral surface area.

In the figure of this problem, the base of the cone is a circle with a <note>radius of 3 cm</note>, and the base area is <note>π(3)^2 = 9π cm²</note>. The lateral surface of the cone, when unfolded, is a sector with a <note>radius of slant height 10 cm</note>, and the arc length of the sector is equal to the <note>circumference of the base 2π(3) = 6π cm</note>. The lateral area is equal to the area of the sector, which is <note>π(3)×10 = 30π cm²</note>. The total surface area of the cone is equal to the base area plus the lateral area, so the total surface area is <note>9π + 30π = 39π cm²</note>.


Question:
Find the height h mm of this closed cylinder if its surface area (S) is $27288$mm2.

Round your answer to the nearest whole number.

Answer: 58

1. Given parameters: The radius of the cylinder r = 43 mm, the surface area S = 27288 mm².

2. According to the formula for the surface area of a cylinder, S = 2πr² + 2πrh, where r is the radius of the cylinder and h is the height of the cylinder.

3. Substitute the given values into the surface area formula: 27288 = 2π(43)² + 2π(43)h.

5. Let the unknown part of the equation 2π(43)h = x, then 2x + 11594.368 = 27288.

8. Solve the equation x = 2πrh for h, yielding h = 15693.632 / (2π(43)).

9. Calculate h ≈ 58.0999. (Round to the nearest integer)

10. Through the above reasoning, the final answer is 58.

Cylinder

A cylinder is a geometric solid consisting of two parallel and congruent circular bases and a curved lateral surface connecting the bases.

In the diagram of this problem, <note>a cylinder consists of two parallel and identical circular bases and a lateral surface</note>. The bases are <note>two identical circles</note>, with <note>radius r = 43 mm</note>, and their <note>centers are on the same line</note>. The lateral surface is a rectangle, and when unfolded, its <note>height is equal to the height h mm of the cylinder</note>, and its <note>width is equal to the circumference of the circle</note>.

Definition of Radius

A radius is a line segment joining the center of a circle to any point on the circumference of the circle.

In the figure of this problem, <note>the length of the line segment from the center of the circle to any point on the circle in the base of the cylinder is 43 mm</note>, therefore <note>this line segment is the radius of the circle</note>.

Definition of Diameter

A diameter is a line segment that passes through the center of a circle and has its endpoints on the circle. It is the longest chord of the circle, with a length equal to twice the radius of the circle.

The original text: <note>The diameter of a circle is a line segment that passes through the center of the circle and has its endpoints on the circle</note>, with a length of 2 times the radius, that is, <note>Diameter = 2r = 2×43 mm = 86 mm</note>.

Surface Area Formula for a Cylinder

The surface area of a cylinder is equal to the sum of the areas of the two bases and the lateral surface area. The total surface area (SA) is given by the formula: SA = 2πr² + 2πrh, where r is the radius of the base circle, and h is the height of the cylinder.

Original text: <note>The surface area of the cylinder S = 27288 mm²</note>, <note>the radius of the base circle r = 43 mm</note>, the height of the cylinder is <note>h mm</note>. Therefore, according to the formula, we have <note>S = 2π(43)² + 2π(43)h</note>.


Question: As shown in the figure, the height of the triangular prism is 8cm. Find the volume of the triangular prism.

Answer: 32 \mathrm{cm}^3

1. Given that the side of the triangular prism is a triangle with a base of 4 cm and a height of 2 cm. According to the triangle area formula, the area of the triangle A can be expressed as A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \text{cm} \times 2 \text{cm} = 4 \text{cm}^2 .

2. According to the volume formula, the volume of the triangular prism V is equal to the base area A multiplied by the height of the prism. The height of the triangular prism is given as 8 cm.

3. Therefore, the volume V = \text{base area} \times \text{prism height} = 4 \text{cm}^2 \times 8 \text{cm} = 32 \text{cm}^3 .

4. Through the above reasoning, the final answer is 32 cubic centimeters.

Area Formula of a Triangle

The base of the triangle is <note>4 cm</note>, and the height is <note>2 cm</note>. According to the area formula of a triangle, the area of the triangle is equal to the base multiplied by the height and then divided by 2, i.e., area = (<note>4 cm</note> * <note>2 cm</note>) / 2 = 4 cm^2.

Volume Formula of Prism

In the diagram of this problem, <note>the base area is 4 cm\(^2\)</note>, <note>the height of the triangular prism is 8 cm</note>, therefore the volume \( V = 4 \text{cm}^2 \times 8 \text{cm} = 32 \text{cm}^3 \).

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

In the figure of this problem, <note>one angle of the triangle is a right angle (90 degrees)</note>, so this triangle is a right triangle. <note>The side lengths of 4 cm and 2 cm are the legs of the right triangle</note>.


Question: Find the surface area of the figure shown. The two marked edges have the same length.

Answer: 120 \mathrm{cm}^2

1. Analyze the figure and find that it consists of a larger cuboid and a smaller cuboid combined.

2. First, calculate the surface area of the larger cuboid. Its dimensions are length 12 cm, width 1 cm, height 1 cm.

3. The formula for the surface area of a cuboid is: 2 (length * width + length * height + width * height). Using this formula, we get: the surface area of the larger cuboid = 2 (12 cm * 1 cm + 12 cm * 1 cm + 1 cm * 1 cm) = 2 (12 + 12 + 1) = 2 * 25 = 50 cm².

4. Then calculate the surface area of the smaller cuboid. Its dimensions are length 4 cm, width 1 cm, height 7 cm.

5. Using the formula for the surface area of a cuboid, we get: the surface area of the smaller cuboid = 2 (4 cm * 1 cm + 4 cm * 7 cm + 1 cm * 7 cm) = 2 (4 + 28 + 7) = 2 * 39 = 78 cm².

6. Next, calculate the overlapping part of the combined figure, which will reduce the total surface area. The overlapping part is the area where the two cuboids connect, and its area is: (4 cm * 1 cm), since it is counted twice for both cuboids, it equals 4 * 2 = 8 cm².

7. Finally, after processing the surface areas of the individual cuboids and subtracting the overlapping part, we get the total surface area of the combined figure: larger cuboid surface area + smaller cuboid surface area - overlapping part = 50 cm² + 78 cm² - 8 cm².

n. The calculation results in a total surface area of 120 cm².

8. Through the above reasoning, the final answer is: 120 cm².

Definition of Rectangular Prism

A rectangular prism is defined as a three-dimensional geometric figure formed by six rectangular faces, where three mutually adjacent faces respectively define the length, width, and height.

The dimensions of the large rectangular prism are <note>12 cm × 1 cm × 1 cm</note>, and the dimensions of the small rectangular prism are <note>4 cm × 1 cm × 7 cm</note>.

Surface Area Formula for Rectangular Prism

The surface area \( S \) of a rectangular prism is given by \( S = 2 \times ( l \times w + w \times h + h \times l ) \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

<note>Surface area of the large rectangular prism</note> = 2 × (<note>12 cm × 1 cm</note> + <note>12 cm × 1 cm</note> + <note>1 cm × 1 cm</note>) = 2 × 25 cm² = 50 cm². <note>Surface area of the small rectangular prism</note> = 2 × (<note>4 cm × 1 cm</note> + <note>4 cm × 7 cm</note> + <note>1 cm × 7 cm</note>) = 2 × 39 cm² = 78 cm².


Question: As shown in the figure, the height of the cylindrical water tank is 3x. The cylindrical water tank is leaking at a constant rate of 3\pi x litres per second (where x is a fixed unknown). If the water tank was initially full, calculate the amount of water remaining after \left(2x+2\right)^2 seconds. Give your simplified answer in expanded form.

Answer: 63\pi x^3+66\pi x^2+15\pi x litres

1. We know that the base of the cylindrical water tank is a circle with a radius of 5x + 3.

2. The formula for the area of a circle is πr^2, so the base area of the cylindrical water tank is π(5x + 3)^2.

3. Given that the height of the cylindrical water tank is 3x, according to the volume formula for a cylinder V = base area * height, the total volume of the tank is π(5x + 3)^2 * 3x.

4. The cylindrical water tank was initially full, so the initial amount of water is π(5x + 3)^2 * 3x liters.

5. Water flows out of the tank at a rate of 3πx liters per second. Given the time is (2x + 2)^2 seconds, then the amount of water lost = rate of flow per second * time lost = 3πx * (2x + 2)^2.

6. Perform the calculation. First, (2x + 2)^2 = 4x^2 + 8x + 4.

7. Substitute the result into step 5, the amount of water lost = 3πx * (4x^2 + 8x + 4).

8. The amount of water lost = 3πx * (4x^2 + 8x + 4) = 12πx^3 + 24πx^2 + 12πx.

9. The remaining amount of water in the tank = initial amount of water - amount of water lost.

10. Initial amount of water V_initial = π(5x + 3)^2 * 3x, simplified to 3πx * ((5x)^2 + 2 * 5x * 3 + 3^2 = (25x^2 + 30x + 9)) = 3πx * (25x^2 + 30x + 9) = 75πx^3 + 90πx^2 + 27πx.

11. Remaining volume = initial amount of water - amount of water lost, which is (75πx^3 + 90πx^2 + 27πx) - (12πx^3 + 24πx^2 + 12πx).

12. Combine: remaining volume = (75πx^3 - 12πx^3) + (90πx^2 - 24πx^2) + (27πx - 12πx).

13. Calculate: remaining volume = 63πx^3 + 66πx^2 + 15πx.

14. Therefore, the remaining amount of water in the tank is 63πx^3 + 66πx^2 + 15πx liters.

Definition of Circle

A circle is the locus of all points in a plane that are at a fixed distance from a given point.

The bottom surface of the cylindrical water tank is a circle, its <note>center is at the bottom center</note>, <note>radius is 5x + 3</note>. All points in the figure that are at a distance equal to <note>5x + 3</note> from the bottom center are on the <note>circular bottom surface</note>.

Area Formula of a Circle

<note>The base of the cylindrical water tank is a circle</note>, its <note>radius is 5x + 3</note>. According to the <note>Area Formula of a Circle</note>, <note>the area A of the circle is equal to pi multiplied by the square of the radius (5x + 3)</note>, which is <note>A = π(5x + 3)²</note>.

Cylinder

In the diagram of this problem, the <note>cylindrical</note> water tank consists of <note>two parallel and identical circular bases</note> and a <note>side</note>. The bases are <note>two completely identical circles</note>, with a <note>radius of 5x + 3</note>, and their <note>centers are on the same line</note>. The side is a <note>rectangle</note>, and when unfolded, its <note>height is equal to the height of the cylinder, 3x</note>, and its <note>width is equal to the circumference, 2π(5x + 3)</note>.

Volume Formula of Cylinder

The volume of a cylinder is equal to the area of its base multiplied by its height. \( V = \pi r^2 h \), where \( r \) is the radius of the base of the cylinder, and \( h \) is the height of the cylinder.

<note>The total volume V of the cylindrical water tank = π(5x + 3)^2 * 3x</note>.


Question:
In the figure shown above, the cylinder has radius of 3. If all the water in the rectangular container is poured into the cylinder, the water level rises from $h$ inches to $(h+x)$ inches. Which of the following is the best approximation of the value of $x$ ?
Choices:
A:3
B:3.4
C:3.8
D:4.2

Answer: D

1. Pour all the water from the rectangular container into the cylinder, and the water level in the cylinder rises from $h$ inches to $(h+x)$ inches. At this time, the volume of water in the rectangular container is equal to the newly added volume of water in the cylinder.

2. Calculate the volume of water in the rectangular container. The container is a rectangular prism with a length of 6 inches, a width of 5 inches, and a water depth of 4 inches. Therefore, the volume of water in the container is $V_{rectangular} = 6 	imes 5 	imes 4 = 120$ cubic inches.

3. Calculate the volume of the added water in the cylinder. The base area of the cylinder is $\pi 	imes (3)^2 = 9\pi$ square inches. So the volume of the added water is $V_{cylinder} = 9\pi 	imes x$ cubic inches.

4. List the equation: the volume of water poured from the rectangular container into the cylinder should be equal, i.e., $120 = 9\pi 	imes x$.

5. Solve the equation $120 = 9\pi x$, and get $x = rac{120}{9\pi}$.

6. Calculate $x = rac{120}{9\pi} \\approx 4.24$, using $\\pi \\approx 3.14$.

7. According to the options, 4.2 is the best approximation of 4.24. Therefore, option D is the correct answer.

8. Based on the above reasoning, the final answer is option D, which is 4.2.

Volume Formula of Rectangular Prism

The volume of a rectangular prism is equal to the product of its length, width, and height.

<note>The length of the rectangular prism is 6 inches, the width is 5 inches, and the height is 4 inches</note>. Therefore, the volume of the rectangular prism V_rectangular = 6 × 5 × 4 = 120 cubic inches.

Area Formula of a Circle

<note>The base of the cylinder is a circle with a radius of 3 inches</note>. Therefore, <note>the base area is π × (3)^2 = 9π square inches</note>.

Volume Formula of Cylinder

<note>The height of the added water is x</note>, so the volume of the added water <note>V_cylinder = 9π × x cubic inches</note>.


Question:
Given the following square pyramid. The length of the base is 10cm.
Find the volume of the square pyramid.

Answer: 200 \mathrm{cm}^3

1. Given a square pyramid, its base is a square with a side length of 10cm.

2. According to the figure, the vertical distance from the tip to the center of the square is the height, which is 6cm.

3. According to the formula for the area of a square, its area is 10cm×10cm=100 square centimeters.

4. According to the formula for the volume of a pyramid, the volume V = (1/3)×base area×height.

5. Substituting the specific values, V = (1/3)×100 square centimeters×6cm.

6. Calculating, we get V = 200 cubic centimeters.

7. Through the above reasoning, the final answer is 200 cubic centimeters.

Definition of Square

A quadrilateral is a square if and only if all four sides are congruent (≅) and all four interior angles are right angles (90°).

In the figure of this problem, <note>the four sides of the base square are all 10 cm long</note>, <note>and all four angles are right angles (90 degrees)</note>, so the base is a square.

Area Formula for Square

The area of a square is equal to the square of its side length, that is, Area = side length × side length.

<note>The side length of the square is 10 cm</note>, so its area is <note>10 cm × 10 cm = 100 square centimeters</note>.

Volume Formula of Pyramid

The volume \( V \) of a pyramid is equal to one third of the product of its base area and its height. Mathematically, this is expressed as: \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \).

The original text: <note>Base area is 100 square centimeters</note>, <note>height is 6 cm</note>. Therefore, the volume V = (1/3) × <note>100 square centimeters</note> × <note>6 cm</note>. According to the formula, the final volume V = 200 cubic centimeters.


Question: As shown in the figure, the height of the base is 14cm. Find the volume of the prism.

Answer: 399 \mathrm{cm}^3

1. In the provided figure, there is a symmetrical triangular prism, where the base is composed of an isosceles trapezoid with the top base length of 14 cm, the bottom base length of 5 cm, and the height of 3 cm.

2. The area of the trapezoid can be calculated using the trapezoid area formula: \( A = \frac{1}{2} \times (a + b) \times h \), where a and b are the lengths of the top and bottom bases, and h is the height of the trapezoid. Therefore: \( A = \frac{1}{2} \times (14 + 5) \times 3 = \frac{1}{2} \times 19 \times 3 = 28.5 \text{ cm}^2 \).

3. According to the information provided at the top, the height of the entire geometric figure, i.e., the triangular prism, is 14 cm.

4. The volume of a prism can be calculated using the formula "V = base area \times height". Therefore: \( V = 28.5 \text{ cm}^2 \times 14 \text{ cm} = 399 \text{ cm}^3 \).

5. Finally, through the detailed derivation above, it is concluded that the volume of the geometric figure is 399 \text{ cm}^3 \.

Definition of Isosceles Trapezoid

A trapezoid is isosceles if and only if its non-parallel sides (legs) are congruent (∅).

<note>The base is an isosceles trapezoid</note>, in which <note>the upper base is 14 cm</note>, <note>the lower base is 5 cm</note>, <note>the height is 3 cm</note>, <note>the two legs are equal</note>.

Definition of Prism

A prism is a polyhedron with two parallel and congruent polygonal bases, and all other faces are parallelograms.

<note>The base of the triangular prism is an isosceles trapezoid</note>, <note>the length of the upper base is 14 cm</note>, <note>the length of the lower base is 5 cm</note>, <note>the height of the trapezoid is 3 cm</note>, <note>the height of the prism (height in solid geometry) is 14 cm</note>.

Trapezoid Area Formula

Given a trapezoid with the lengths of its two parallel sides as \(a\) and \(b\), and the height between these sides as \(h\), the area of the trapezoid is given by \(\frac{(a + b) \cdot h}{2}\).

<note>The upper base of the trapezoid is 14 cm</note>, <note>the lower base is 5 cm</note>, <note>the height is 3 cm</note>, so the area is calculated as: \( A = \frac{1}{2} \times (14 + 5) \times 3 = 28.5 \text{ cm}^2 \).

Volume Formula of Prism

The volume of a prism is equal to the base area multiplied by the height.

<note>The base area is 28.5 cm²</note>, <note>the height of the prism is 14 cm</note>, so the volume is calculated as: \( V = 28.5 \text{ cm}^2 \times 14 \text{ cm} = 399 \text{ cm}^3 \).


Question: The height of the right pyramid is 7mm. Find the exact volume of the right pyramid pictured.

Answer: \frac{847}{3} \mathrm{mm}^3

1. Given that the height of the regular pyramid is 7mm, according to the geometric feature diagram, we can determine that the base of the regular quadrilateral pyramid is a square.

2. According to the properties of the square, its sides are all equal. Since the length of the base edge is one of the corresponding parts of the square's side length, given that each edge is 11mm, the side length of the square base is 11mm.

3. When calculating the area of the square base, according to the area formula of the square A = a^2, where a is the side length of the square. This can be confirmed based on the diagram on the base. Substituting the side length of the square a = 11mm: A = 11 * 11 = 121mm^2.

4. To calculate the volume of the regular pyramid, use the formula V = (1/3) * A * h, where A represents the base area, and h is the vertical height. Substituting the previously calculated base area A = 121mm^2 and vertical height h = 7mm, we get:

5. Calculating the volume V = (1/3) * 121 * 7 = 847/3 mm^3.

6. Through the above reasoning, the exact volume of the regular pyramid is: 847/3mm³.

Definition of Square

<note>The base is a square</note>, with <note>four equal sides each measuring 11 mm</note>, and <note>all four interior angles are right angles (90 degrees)</note>.

Definition of Regular Pyramid

A regular pyramid is defined as a pyramid in which the base is a regular polygon, and each of the lateral faces is an identical isosceles triangle.

<note>The base is a square with a side length of 11mm</note>, <note>the height of the regular pyramid is 7mm</note>.

Area Formula for Square

<note>The side length of the square base is 11mm</note>, the base area <note>A = 11mm * 11mm = 121 mm²</note>.

Volume Formula of Pyramid

<note>Base area A = 121 mm²</note> and <note>height h = 7mm</note>, substituting into the formula yields the volume <note>V = (1/3) * 121 * 7 = 847/3 mm^3</note>, i.e., <note>rac{847}{3} mm³</note>.


Question: As shown in the figure, the height of the rectangular prism is 3cm. Find the volume of the rectangular prism.

Answer: 48 \mathrm{cm}^3

1. Given that the rectangular prism has a length of 8 cm, a width of 2 cm, and a height of 3 cm.

2. According to the volume formula of a rectangular prism: Volume = Length * Width * Height.

3. Insert the given data: Volume = 8 cm * 2 cm * 3 cm.

4. Calculate the result: Volume = 48 cubic cm.

5. Through the above reasoning, the final answer is 48 cubic cm.

Definition of Rectangular Prism

In the figure of this problem, <note>the six faces of the geometric body are all rectangles</note>, <note>its length is 8 cm</note>, <note>width is 2 cm</note>, <note>height is 3 cm</note>, so this is a rectangular prism.

Volume Formula of Rectangular Prism

Original: <note>The length of the rectangular prism is 8 cm, the width is 2 cm, and the height is 3 cm</note>, according to the definition, <note>Volume = 8 cm * 2 cm * 3 cm</note>.

Formula for the Area of a Rectangle

The area of a rectangle is equal to its length multiplied by its width.

In the <note>base of the cuboid (i.e., a rectangle)</note>, <note>the sides 8 cm and 2 cm are the length and width of the rectangle</note>, so the <note>area of the rectangle</note> = 8 cm * 2 cm = 16 square cm.


Question:
A cylinder has a surface area of 54105 \mathrm{mm}^2.

What must the height h mm of the cylinder be?

Round your answer to the nearest whole number.

Answer: 30

1. From the given information, we know that the total surface area of the cylinder is 54105 	ext{mm}^2, the radius of the base of the cylinder is 79 	ext{mm}, and we need to find the height h mm.

2. The formula for the surface area of a cylinder is: Surface Area = 2	ext{π}rh + 2	ext{π}r^2, where r is the radius of the base and h is the height.

3. Substituting the given radius r = 79 and the given surface area value, the equation is: 54105 = 2 * 	ext{π} * 79 * h + 2 * 	ext{π} * 79^2.

4. Calculate the base area term: 2 * 	ext{π} * r^2 = 2 * 	ext{π} * (79)^2 = 2 * 	ext{π} * 6241 = 12482	ext{π}.

5. Substitute the base area term into the equation, we get: 54105 = 2 * 	ext{π} * 79 * h + 12482	ext{π}.

6. Rearrange the equation: 54105 - 12482	ext{π} = 2 * 	ext{π} * 79 * h.

7. Remove the 	ext{π} term: 54105 - 12482	ext{π} ≈ 54105 - 39193.48 = 14911.52.

8. Calculate h: 14911.52 = 2 * 	ext{π} * 79 * h, then h = rac{14911.52}{2 * 	ext{π} * 79} = rac{14911.52}{496.12}.

9. Solve for h ≈ 30.06. According to the problem, we need to round to the nearest integer, so h = 30.

10. Through the above reasoning, the final answer is <30>.

Cylinder

A cylinder is a geometric solid consisting of two parallel and congruent circular bases and a curved lateral surface connecting the bases.

In the diagram of this problem, the <note>cylinder</note> consists of two parallel and identical <note>circular bases</note> and a <note>lateral surface</note>. The <note>bases are two identical circles</note> with a radius of <note>79 mm</note>, and <note>their centers are on the same line</note>. The <note>lateral surface is a rectangle, and when unfolded, its height is equal to the height h mm of the cylinder, and its width is equal to the circumference of the circle</note>.

Definition of Circle

The <note>center of the circle</note> is a fixed point, <note>radius is 79 mm</note>. In the figure, <note>all points that are 79 mm away from the center of the circle</note> are on the circle at the base of the cylinder.

Surface Area Formula for a Cylinder

The surface area of a cylinder is equal to the sum of the areas of the two bases and the lateral surface area. The total surface area (SA) is given by the formula: SA = 2πr² + 2πrh, where r is the radius of the base circle, and h is the height of the cylinder.

The original text: <note>The surface area of the cylinder is 54105 mm²</note>, <note>the base radius r is 79 mm</note>, and the height h is unknown. By <note>substituting into the formula 2πr² + 2πrh</note>, the required <note>height h</note> can be obtained.


Question: Find the surface area of the triangular prism shown. The height is 20 cm.

Answer: 528 \mathrm{cm}^2

1. Given that the hexahedron is a triangular prism, by observation, we can see that its base is a right triangle.

2. The lengths of the two legs of the right triangle are 6 cm and 8 cm respectively (these two sides are marked in the figure).

3. To find the area of the right triangle, according to the triangle area formula S=1/2 * base * height, the area = 1/2 * 6 cm * 8 cm = 24 square cm.

4. The top of the triangular prism and the face parallel to the base are formed by the same right triangle.

5. The total surface area of the hexahedron is equal to the sum of the areas of its two triangular bases plus the areas of the three lateral faces.

6. The combined area of the two bases is 2 * 24 square cm = 48 square cm.

7. Each of the three lateral faces contains a rectangular side, with the lengths being the three sides of the base (including the hypotenuse) and the height of the triangular prism, which is 20 cm long.

8. The areas of these three lateral faces are:

   (The side formed by the leg of 6 cm and the height of 20 cm, area is: 6 cm * 20 cm = 120 square cm)

   (The side formed by the leg of 8 cm and the height of 20 cm, area is: 8 cm * 20 cm = 160 square cm)

   (The side formed by the hypotenuse of 10 cm and the height of 20 cm, area is: 10 cm * 20 cm = 200 square cm)

9. Adding all the lateral areas together, we get the total lateral area as 120 + 160 + 200 = 480 square cm.

10. Finally, adding the areas of the two bases and the three lateral faces, we get the total surface area of the triangular prism.

11. The total surface area is 48 square cm (base area) + 480 square cm (lateral area) = 528 square cm.

12. Through the above reasoning, the final answer is that the surface area of the triangular prism is 528 square cm.

Definition of Triangular Prism

A triangular prism is a type of hexahedron that is formed by two parallel and congruent triangular bases and three rectangular lateral faces.

<note>In the diagram of this problem, the entire solid figure is a triangular prism.</note> Its two bases are <note>plane triangles</note> (<note>one base is marked with sides 6 cm and 8 cm in the diagram</note>), and its three lateral faces are <note>rectangles</note>, with a height of <note>20 cm</note>.

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

In the diagram of this problem, the base is a right triangle. <note>The legs are 6 cm and 8 cm respectively</note>, <note>the right angle is located at their intersection</note>. <note>The hypotenuse is 10 cm</note>.

Definition of Rectangle

A quadrilateral is called a rectangle if and only if each of its interior angles is a right angle (90°) and its opposite sides are both parallel (∥) and congruent (≅).

In the figure of this problem, the three lateral faces of the triangular prism are <note>rectangles</note>. Each rectangle has interior angles that are <note>right angles (90 degrees)</note>, and <note>opposite sides are parallel and equal in length</note>. The side lengths of the rectangles are <note>6 cm, 8 cm, and 10 cm</note> and <note>height 20 cm</note>.

Area Formula of a Triangle

The area of any triangle is equal to its base multiplied by its height, divided by 2.

In the figure of this problem, <note>in the right triangle of the base, the side 6 cm is the base, and the side 8 cm is the height</note>. According to the area formula of a triangle, the area of the right triangle of the base is equal to <note>the base 6 cm multiplied by the height 8 cm and then divided by 2</note>, that is, Area = (6 cm * 8 cm) / 2 = 24 square centimeters.

Formula for the Area of a Rectangle

<note>Side 6 cm and side 20 cm are the length and width of the rectangle</note>, so the area of the rectangle = 6 cm * 20 cm = 120 square centimeters;
<note>Side 8 cm and side 20 cm are the length and width of the rectangle</note>, so the area of the rectangle = 8 cm * 20 cm = 160 square centimeters;
<note>Side 10 cm and side 20 cm are the length and width of the rectangle</note>, so the area of the rectangle = 10 cm * 20 cm = 200 square centimeters.

Surface Area Formula for a Triangular Prism

The surface area of a triangular prism is equal to the sum of the areas of its two triangular bases plus the sum of the areas of its three rectangular lateral faces, that is, Surface Area = 2 * Base Area + Lateral Area.

<note>Surface Area of a Triangular Prism</note> = 2 * <note>Base Area</note> + <note>Sum of Lateral Areas</note> = 2 * 24 square centimeters + 120 square centimeters + 160 square centimeters + 200 square centimeters = 48 square centimeters + 480 square centimeters = 528 square centimeters.


Question:
Which two triangles are congruent by the ASA Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △DEF≅△UWV

1. Observe the given image and find that triangles DEF and UVW are marked with equal angles and sides.

2. Given ∠D≅∠U, it can be seen in the image that these two angles have the same markings.

3. Given DE ≅ UW, it can be seen in the image that side DE and side UW have the same tick marks, indicating equal lengths.

4. Similarly, it can be observed that ∠E≅∠W, because these two angles also have the same markings.

5. Using the Angle-Side-Angle (ASA) criterion for triangle congruence: If two angles and the side between them in one triangle are respectively congruent to two angles and the side between them in another triangle, then the triangles are congruent. In this specific problem, triangles DEF and UVW satisfy this condition.

6. By comparing the known corresponding angles and sides, it is concluded that triangles DEF and UVW are congruent.

7. Under the ASA condition, since ∠D≅∠U and ∠E≅∠W, vertex D corresponds to U, vertex E corresponds to W, so vertex F corresponds to V.

8. Therefore, we can finally confirm: △DEF≅△UWV.

If two angles and the included side of one triangle are respectively equal to two angles and the included side of another triangle, then the two triangles are congruent.

In the diagram for this problem, in triangles DEF and UVW, <note>∠EDF≅∠WUV</note>, <note>∠DEF≅∠UWV</note>, and <note>side DE≅side UW</note>. Since the two angles and the included side of these two triangles are equal respectively, according to the Angle-Side-Angle (ASA) Criterion for Congruence of Triangles, it can be concluded that <note>triangle DEF is congruent to triangle UVW</note>.


Question: After bisecting an angle measuring 10°, Olivia has two angles. What is the measure of each new angle? $\Box$ °

Answer:
| --- |
| 5° |

1. Let the bisected 10° angle be ∠ABC, and BD be the angle bisector.

2. According to the definition of the angle bisector, when a ray bisects an angle, it divides the angle into two equal angles, i.e., ∠ABD = ∠DBC.

3. Since ∠ABC is bisected into two equal angles and the total degree is 10°, each new angle measure is ∠ABC/2=10°/2=5°.

4. Therefore, according to the above reasoning process, each new angle measure is 5°.

Definition of Angle Bisector

An angle bisector is a line that originates from the vertex of an angle and divides the angle into two congruent angles.

<note>The vertex of the angle is point B</note>, from <note>point B, a line BD is drawn</note> to divide <note>the angle of 10° into two equal angles</note>, each being 5°. Therefore, <note>line BD is the angle bisector of the 10° angle</note>.


Question:
The following is a right pyramid on a square base with side length 16cm. A right pyramid has its apex aligned directly above the centre of its base.

Find the length of VW, the perpendicular height of the pyramid correct to two decimal places.

Answer: 23.41

1. Given that the base of the square is ABCD, the center of the base is point W.

2. In ABCD, since the base is a square with a side length of 16 cm, the diagonal AC can be calculated using the Pythagorean theorem.

4. According to the properties of the square's diagonal, AW = WC = 1/2 AC = (√512)/2 = 8√2. That is, AW = 8√2.

5. In △VAW, given that VW is the vertical height, VW ⊥ AC. According to the definition of a right triangle, ∠VWA is a right angle. Given VA = 26 cm, according to the Pythagorean theorem, VA = √(VW^2 + AW^2).

6. Substituting AW and VA, we get 26^2 = (8√2)^2 + VW^2.

8. Solving the equation, we get VW^2 = 676 - 128 = 548.

9. Taking the square root of VW: VW = √548.

10. Calculating VW, we get VW ≈ 23.41 (rounded to two decimal places).

11. Through logical verification, the final answer is obtained.

Definition of Square

A quadrilateral is a square if and only if all four sides are congruent (≅) and all four interior angles are right angles (90°).

<note>Side AB, side BC, side CD, and side DA are equal</note>, and <note>angle DAB, angle ABC, angle BCD, and angle CDA are all right angles (90 degrees)</note>, so <note>ABCD is a square</note>.

Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs.

In the figure of this problem, <note>triangle VAW is a right triangle</note>, <note>∠VWA is a right angle (90 degrees)</note>, <note>sides VW and AW are the legs</note>, <note>side VA is the hypotenuse</note>, so according to the <note>Pythagorean Theorem</note>, <note>VA² = VW² + AW²</note>.

Definition of Right Triangle

In <note>triangle VAW</note>, <note>angle VWA is a right angle (90 degrees)</note>, therefore <note>triangle VAW is a right triangle</note>.

Properties of Diagonals in a Square

The diagonals of a square are the line segments that connect opposite vertices. The diagonals of a square are equal in length, and they bisect each other perpendicularly.

In the given figure, in square ABCD, the diagonals AC and BD are line segments connecting opposite corners. According to the properties of diagonals in a square, AC and BD are equal, and AC and BD bisect each other perpendicularly, therefore, AW = WC.


Question:
Which two triangles are congruent by the AAS Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △HIJ≅△XWY

1. Given ∠H ≅ ∠X and ∠J ≅ ∠Y, according to the congruent triangles theorem (AAS), if two triangles have two angles and the non-included side respectively equal, then the two triangles are congruent.

2. Observe the figure, look for the two angles and the corresponding non-included side that correspond to ∠H and ∠J, and find that in △WXY, ∠X and ∠Y correspond to ∠H and ∠J respectively.

3. Check the non-included sides of △HIJ and △WXY, side IJ ≅ side WY, which meets the requirements of the congruent triangles theorem (AAS).

4. According to the above steps, the two angles and the non-included side are respectively equal, so according to the congruent triangles theorem (AAS), △HIJ ≅ △XWY.

5. To write the congruence statement, according to ∠H corresponding to ∠X, ∠J corresponding to ∠Y, and side IJ corresponding to side WY, the congruence relationship can be further confirmed as △HIJ ≅ △XWY.

Congruence Theorem for Triangles (AAS)

Two triangles are congruent if two angles and the included side of one triangle are respectively equal to two angles and the included side of the other triangle.

<note>Angle X is equal to angle H</note>, <note>angle Y is equal to angle J</note>, <note>side WY is equal to side JI, and this side is not between the two known angles</note>. Therefore, according to the Congruence Theorem for Triangles (AAS), <note>triangle HIJ is congruent to triangle WXY</note>.


Question:
Which two triangles are congruent by the AAS Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △IJK≅△EDC

1. Given: In △IJK, ∠I ≅ ∠E and ∠K ≅ ∠C, and side IJ ≅ side DE.

2. According to the congruent triangles theorem (AAS): If two triangles have one pair of corresponding angles equal respectively and one pair of corresponding sides not connected to these angles equal, then the two triangles are congruent.

3. In this problem, we can see that △IJK and △CDE satisfy the congruent triangles theorem (AAS), because: ∠I ≅ ∠E, ∠K ≅ ∠C, and side IJ ≅ side DE.

4. After the above reasoning, the final answer is △IJK ≅ △EDC.

Naming of Angles

An angle can be named using three points, with the vertex point located in the middle, or it can be named solely by the vertex.

<note>∠I</note> represents angle <note>KIJ</note>, <note>∠E</note> represents angle <note>CED</note>, <note>∠K</note> represents angle <note>IKJ</note>, <note>∠C</note> represents angle <note>DCE</note>, <note>∠J</note> represents angle <note>IJK</note>, and <note>∠D</note> represents angle <note>EDC</note>.

Congruence Theorem for Triangles (AAS)

<note>△IJK</note> and <note>△CDE</note> satisfy the AAS theorem:
1. <note>∠I ≅ ∠E</note>
2. <note>∠K ≅ ∠C</note>
3. <note>side IJ ≅ side DE</note>

Therefore, it is concluded that <note>△IJK ≅ △EDC</note>.


Question: The height is 13 cm. Find L. Give your answer rounded down to the nearest cm.

Answer: 14

1. Given that the height of the cylinder is 13 cm and the radius is 3 cm, the problem requires finding the length of L and rounding the result to the nearest centimeter.

2. The base of the cylinder is a circle, and the diameter of the base is the distance between any two opposite sides of the base.

3. The diagonal L of the cylinder at the position of the diameter on the circumference of the top and bottom circles is formed by the height and the diameter of the base.

4. It can be calculated using the Pythagorean theorem, where L is the hypotenuse of the right triangle, and the other two sides are the diameter of the circle and the height of the cylinder.

5. The Pythagorean theorem is expressed as a² + b² = c², here it is the square of the diameter 6 cm plus the square of the height 13 cm equals the square of the diagonal L, i.e., 6² + 13² = L².

6. The calculation yields, 36 + 169 = L², 205 = L².

7. Finally, calculating L, we get L = √205 ≈ 14.3178.

8. Rounding down, L is approximately 14 cm.

9. Through the above reasoning, the final answer is 14.

Cylinder

<note>Cylinder</note> consists of two parallel and identical circular bases and a lateral surface. The bases are two identical circles at the top and bottom, with a <note>radius of 3 cm</note>, <note>diameter of 6 cm</note>, and their centers are aligned on the same line. The lateral surface is a rectangle, and when unfolded, its <note>height equals the cylinder's height of 13 cm</note>, and its <note>width equals the circumference of the circle</note>.

Definition of Diameter

A diameter is a line segment that passes through the center of a circle and has its endpoints on the circle. It is the longest chord of the circle, with a length equal to twice the radius of the circle.

<note>The radius of the base circle of the cylinder is 3 cm</note>, therefore <note>the diameter is 6 cm</note>. According to the definition, <note>the diameter is the line segment that passes through the center of the circle and has both ends on the circle</note>, it is the longest chord of the circle, with a length of 2 times the radius. Therefore, in the definition of this problem, <note>the diameter is 6 cm</note>.

Definition of Right Triangle

The height of the cylinder (<note>13 cm</note>) and the diameter of the base (<note>6 cm</note>) form a <note>right triangle</note>. The two legs of this right triangle are <note>height 13 cm</note> and <note>diameter 6 cm</note>, and the <note>hypotenuse is L</note>.

Pythagorean Theorem

<note>The height of the cylinder, 13 cm, and the diameter of the base circle, 6 cm, form the two perpendicular sides of a right triangle</note>. According to the Pythagorean Theorem, <note>the square of the hypotenuse L is equal to the square of the diameter (6 cm) plus the square of the height (13 cm)</note>. Mathematically expressed as <note>6² + 13² = L²</note>, which calculates to <note>L = √205 ≈ 14.3178 cm</note>, finally rounded to <note>14 cm</note>.


Question:
In the cylindrical tube shown above, the circumference of the circular base is 32 . If the tube is cut along $\overline{A B}$ and laid out flat to make a rectangle, what is the length of $\overline{A C}$ to the nearest whole number?
Choices:
A:24
B:30
C:34
D:38

Answer: C

1. Given that the base of the cylinder is circular, and the circumference of the base is 32.

2. If the cylinder is cut along the line \\overline{AB} and unfolded into a plane, the circumference of the base will become one side of the unfolded rectangle, and the line \\overline{AB} will become the other side.

3. The length of the unfolded rectangle is the circumference 32, and the width is the original height of the cylinder \\overline{AB} = 30.

4. Let the unfolded rectangle be A'ABB', \\overline{AC} is the diagonal of the rectangle ABCD.

5. Using the Pythagorean theorem, \\overline{AC} can be calculated. The Pythagorean theorem states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.

6. In the rectangle ABCD, \\overline{AC} is the diagonal, and BC is half of the circumference, so by the formula \\$AC = \\sqrt{16^2 + 30^2}\\$.

7. Calculation: \\$AC = \\sqrt{16^2 + 30^2} = \\sqrt{256 + 900} = \\sqrt{1156}\\$.

8. The square root calculation of \\sqrt{1156} gives 34.

9. The length of \\overline{AC} is 34.

Definition of Rectangle

In the figure of this problem, the unfolded rectangle A'ABB' has a length of 32, which is the circumference of the cylinder's base, and a width equal to the height of the cylinder <note>AB = 30</note>. The quadrilateral <note>ABCD is a rectangle</note>, with its interior angles <note>∠DAB, ∠ABC, ∠BCD, ∠CDA all being right angles (90 degrees)</note>, and the side <note>AD is parallel and equal in length to side BC</note>, while the side <note>AB is parallel and equal in length to side CD</note>. In rectangle ABCD, <note>BC is half of the circumference</note>.

Pythagorean Theorem

<note>After unfolding, the two perpendicular sides of the rectangle are 16 in length and 30 in width</note>, <note>the diagonal AC is the hypotenuse</note>. Using the Pythagorean Theorem to calculate, <note>the length of AC is \sqrt{16^2 + 30^2} = \sqrt{1156} = 34</note>.


Question: If an angle measuring 60° is bisected to form two new angles, what is the measure of each new angle? $\Box$ °

Answer:
| --- |
| 30° |

1. Given that the measure of ∠ABC is 60°.

2. According to the definition of an angle bisector, the angle bisector divides an angle into two equal angles.

3. Based on the definition of an angle bisector, in the figure, we introduce a ray BD such that it is the angle bisector of ∠ABC.

4. Therefore, according to the definition of an angle bisector, we have ∠ABD = ∠DBC.

5. Since ∠ABD + ∠DBC = ∠ABC and ∠ABD = ∠DBC, according to the definition of an angle bisector, we get 2 * ∠ABD = ∠ABC.

6. Substituting the given ∠ABC = 60°, we calculate: 2 * ∠ABD = 60°.

7. Solving the equation, we get ∠ABD = 60° / 2.

8. The calculation gives ∠ABD = 30°.

9. Similarly, ∠DBC = 30°.

10. Through the above reasoning, the final answer is 30°.

Definition of Angle Bisector

<note>In the diagram of this problem, it is known that ∠ABC measures 60°</note>. Introduce a ray BD such that it is the <note>angle bisector of ∠ABC</note>. According to the definition of angle bisector, <note>∠ABD and ∠DBC are divided into two equal angles by the angle bisector BD</note>. Therefore, <note>∠ABD = ∠DBC</note>.


Question: An angle is bisected, forming two new angles. Each new angle has a measure of 72°. What was the measure of the original angle? $\Box$ °

Answer:
| --- |
| 144° |

1. Given that an angle is bisected to form two new angles, each measuring 72°.

 2. According to the definition of angle bisector, the original angle is divided into two equal angles. Therefore, the sum of the measures of the two new angles equals the measure of the original angle.

 3. Since both new angles measure 72°, the measure of the original angle can be expressed as: 72° + 72°.

 4. From this calculation, the measure of the original angle is 72° + 72° = 144°.

 5. Through the above reasoning, the final answer is 144°.

In the figure of this problem, <note>the original angle is bisected, forming two new angles</note>, <note>the degree of the original angle is 72° + 72° = 144°</note>.


Question:
Which two triangles are congruent by the AAS Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △TUV≅△GFH

1. According to the problem, find two triangles such that each triangle has two pairs of corresponding equal angles and one pair of equal non-included sides.

2. Observe the figure, triangles TUV, FGH, and EDC. It is found that: ∠V = ∠H, ∠T = ∠G. At this point, point T corresponds to point G, point V corresponds to point H.

3. Side UV and side FH are non-included sides, side UV = side FH.

4. Using the AAS theorem, side UV and side FH are in the same position, and the two corresponding angles ∠V and ∠H and ∠T and ∠G are equal, therefore △TUV and △GFH are congruent by the AAS theorem.

5. Through the above reasoning, the final answer is △TUV ≅ △GFH.

Congruence Theorem for Triangles (AAS)

In the figure of this problem, <note>in triangles TUV and GFH, angle V = angle H, angle T = angle G, and side UV = side FH</note>. Therefore, according to the <note>AAS theorem</note>, we can conclude that <note>△TUV ≅ △GFH</note>.

Definition of Congruent Triangles

Two triangles are congruent if and only if their corresponding sides are equal and their corresponding angles are equal.

<note>By the AAS theorem</note>, it can be proven that <note>triangle TUV</note> and <note>triangle GFH are congruent</note>. According to the markings in the diagram, <note>angle V = angle H</note>, <note>angle T = angle G</note>, and <note>side UV = side FH</note>. <note>Therefore △TUV ≅ △GFH</note>.


Question:
Consider the following figure. The height of the prism is 16cm.

Use Pythagoras' Theorem to find the unknown height $x$ in the triangular prism shown.

Answer: \$x=8 \mathrm{~cm}\$

1. Let the vertices of the triangle be A, and the endpoints of the base be B and C (B on the left side). The perpendicular segment AD is x, and it is known that the segment AC is 10 cm, BC is 12 cm. Use the Pythagorean theorem for analysis. According to the Pythagorean theorem, the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.

2. From the given conditions, in triangle ABC, AB = AC, which satisfies the definition of an isosceles triangle. Triangle ABC is an isosceles triangle. According to the theorem that the altitude, median, and angle bisector coincide in an isosceles triangle, when a perpendicular is drawn from point D to point A, AD is also the median of the side BC of the isosceles triangle. D is the midpoint of side BC, CD = 12/2 = 6.

3. In the right triangle ABD (with one angle being 90°, satisfying the definition of a right triangle), using the Pythagorean theorem, we get AD^2 + BD^2 = AB^2. Therefore, we have x^2 + 6^2 = 10^2.

4. Substitute the known values into the equation and simplify: x^2 = 10^2 - 6^2.

5. Substitute the values and calculate, x^2 = 100 - 36.

6. Simplify the equation and solve for x, x = 8.

7. Through the above reasoning and analysis, finally x = 8 cm.

Definition of Right Triangle

<note>In triangle ABD, angle ∠ADB is a right angle (90 degrees)</note>, therefore triangle ABD is a right triangle. The sides <note>AD and BD are the legs</note>, and the side <note>AB is the hypotenuse</note>.

Midpoint of a Line Segment

A midpoint of a line segment is the point that divides the line segment into two equal parts.

<note>The midpoint of line segment BC is point D</note>. According to the definition of the midpoint of a line segment, <note>point D divides line segment BC into two equal parts</note>, that is, <note>the lengths of line segment BD and line segment DC are equal</note>. That is, <note>BD = DC = BC/2 = 12 cm / 2 = 6 cm</note>.

Pythagorean Theorem

In the diagram of this problem, according to the Pythagorean Theorem, in a right triangle, <note>the sum of the squares of the two legs is equal to the square of the hypotenuse</note>. Let the vertex of the right angle of the right triangle be <note>A</note>, <note>the two endpoints of the base be B and C</note>, <note>the perpendicular segment AD be x</note>, and it is known that <note>segment AC is equal to 10cm</note>, <note>BC is equal to 12cm</note>, using the Pythagorean Theorem for analysis. According to the Pythagorean Theorem, <note>the sum of the squares of the two legs of the right triangle is equal to the square of the hypotenuse</note>. It is known that <note>AB=10cm</note>, <note>BD=6cm</note>, so <note>AD^2 + 6^2 = 10^2</note>.

Coincidence Theorem of Altitude, Median, and Angle Bisector in Isosceles Triangle

In an isosceles triangle, the angle bisector of the vertex angle not only bisects the vertex angle but also bisects the base and is perpendicular to the base.

In the diagram of this problem, in the isosceles triangle ABC, the vertex angle is <note>angle A</note>, and the base is <note>side BC</note>. The angle bisector of the vertex angle <note>AD</note> not only bisects <note>vertex angle A</note> but also bisects <note>base BC</note>, making <note>BD = DC</note>, and is perpendicular to <note>base BC</note>, forming a <note>right angle ADB (90 degrees)</note>. Therefore, <note>segment AD</note> is both the angle bisector of the vertex angle and the median and altitude of the base.

Definition of Isosceles Triangle

A triangle is an isosceles triangle if and only if it has at least two congruent (≅) sides.

<note>Side AB is equal to side AC</note>, therefore <note>triangle ABC is an isosceles triangle</note>.


Question:
Which two triangles are congruent by the AAS Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △HIJ≅△CAB

1. According to the problem statement, the AAS (Angle-Angle-Side) theorem states that when two triangles have two corresponding angles equal respectively, and the non-adjacent side to these two angles also equal, the two triangles are congruent.

2. We need to find two sets of triangles in the figure that satisfy the conditions of the AAS theorem: having two pairs of corresponding equal angles and one pair of corresponding non-included sides equal.

3. In △HIJ, according to the markings in the figure, ∠I ≅ ∠A, ∠H ≅ ∠C. Additionally, side IJ ≅ side AB.

4. Therefore, △HIJ and △ABC have two pairs of corresponding angles equal (∠I ≅ ∠A, ∠H ≅ ∠C), and the non-adjacent side IJ and side AB are also equal.

5. According to the AAS (Angle-Angle-Side) theorem, this means △HIJ and △ABC are congruent.

6. When matching corresponding vertices, since ∠I ≅ ∠A and ∠H ≅ ∠C, I corresponds to A, H corresponds to C, and finally J corresponds to B.

7. Therefore, the order of corresponding vertices in the congruent triangles is △HIJ ≅ △CAB.

8. After the above reasoning, the final answer is △HIJ ≅ △CAB.

Definition of Congruent Triangles

In the figure of this problem, <note>△HIJ and △CAB are congruent</note> because they satisfy the AAS theorem. The <note>corresponding sides IJ and AB of the two triangles are equal</note>, the <note>corresponding angles ∠HIJ and ∠CAB of the two triangles are equal</note>, and <note>∠HJI and ∠CBA are equal</note>.

Congruence Theorem for Triangles (AAS)

In the figure of this problem, △HIJ and △CAB, <note>∠I =∠A</note>, <note>∠H =∠C</note>, <note>side IJ = side AB</note>. Since the two triangles have two corresponding angles and the side opposite one of these angles equal, according to the Congruence Theorem for Triangles (AAS), we can conclude that <note>△HIJ is congruent to △CAB</note>.


Question: An angle is bisected, forming two new angles. If the original angle had a measure of 90°, what is the measure of each new angle? $\Box$ °

Answer:
| --- |
| 45° |

1. Given an angle, its measure is 90°.

2. The problem states that this angle is bisected, that is, divided into two equal angles by a ray.

3. According to the definition of an angle bisector, an angle bisector is a ray that originates from the vertex of the angle and divides the angle into two equal angles.

4. Let the original angle be ∠AOB, and the two new angles formed after bisection be ∠AOC and ∠COB.

5. According to the definition of an angle bisector, ∠AOC = ∠COB.

6. Since the measure of the original angle ∠AOB is 90°, and ∠AOB is bisected, the sum of ∠AOC and ∠COB is 90°.

7. Using an equation to solve: if we let ∠AOC = ∠COB = x, then x + x = 90°.

8. Solving this equation gives: 2x = 90°, therefore x = 90° ÷ 2 = 45°.

9. Through the above reasoning, we find that ∠AOC and ∠COB are both 45°.

10. Therefore, the measure of each new angle is 45°.

Definition of Angle Bisector

<note>The vertex of the angle is point O</note>, a ray is drawn from <note>point O</note> to divide the angle into <note>two equal angles</note>, that is, <note>each of the new angles measures 45°</note>. Therefore, <note>this ray is the angle bisector</note>.


Question:
Which two triangles are congruent by the ASA Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △TUV≅△RSQ

1. According to the given figure information, we notice the following known conditions: ∠T and ∠R are congruent, so ∠T = ∠R.

2. Observing the side lengths, we see that side TV and side QR are of equal length, so side TV = side QR.

3. Continuing to observe, we notice that ∠V and ∠Q are also congruent, so ∠V = ∠Q.

4. According to the Angle-Side-Angle (ASA) Congruence Theorem, if two triangles have two angles and the side between those angles respectively congruent, then the two triangles are congruent. Specifically in the problem: ∠T of △TUV = ∠R of △QRS, side TV = side QR, and ∠V = ∠Q.

5. Therefore, by the Angle-Side-Angle Congruence Theorem, we can deduce that △TUV ≅ △RSQ.

6. Through the above reasoning, we finally conclude that △TUV ≅ △RSQ.

In the figure of this problem, in triangles TUV and QRS, <note>angle T is equal to angle R</note>, <note>angle V is equal to angle Q</note>, and <note>side TV is equal to side QR</note>. Since these two triangles have <note>two angles and the included side respectively equal</note>, according to the Angle-Side-Angle (ASA) Criterion for Congruence of Triangles, it can be concluded that <note>triangle TUV is congruent to triangle QRS</note>.


Question: If an angle is bisected to form two new 11° angles, what was the measure of the original angle? $\Box$ °

Answer:
| --- |
| 22° |

1. Let the original angle be ∠ABC, where BD is the angle bisector, dividing ∠ABC into ∠ABD and ∠DBC.

2. According to the given conditions, ∠ABD = 11° and ∠DBC = 11°.

3. According to the definition of the angle bisector, ∠ABD = ∠DBC.

4. Substituting the given conditions ∠ABD = 11° and ∠DBC = 11°, we get ∠ABD = ∠DBC = 11°.

5. Since ∠ABD + ∠DBC = ∠ABC, substituting the known degrees of ∠ABD and ∠DBC, we get ∠ABC = 11° + 11°.

6. Calculating, we get ∠ABC = 22°.

7. Through the above reasoning, the final answer is 22°.

<note>The vertex of the angle is point B</note>, a ray is drawn from <note>point B</note> that divides the angle into two equal angles, namely <note>two 11° angles</note>. Therefore, this ray is the angle bisector of the angle.


Question: If an angle is bisected to form two new 5° angles, what was the measure of the original angle? $\Box$ °

Answer:
| --- |
| 10° |

1. Let the original angle be ∠ABC, and an angle bisector BD divides ∠ABC into two equal angles.

2. It is known that both ∠ABD and ∠DBC are 5°.

3. Therefore, ∠ABC = ∠ABD + ∠DBC.

4. Thus, ∠ABC = 5° + 5° = 10°.

5. Through the above reasoning, the final answer is 10°.

In the figure of this problem, <note>the vertex of the angle is point B</note>, a line BD is drawn from <note>point B</note>, <note>this line divides angle ABC into two equal angles</note>, namely <note>angle ABD and angle DBC are equal</note>. Therefore, <note>line BD is the angle bisector of angle ABC</note>.


Question: As shown in the figure, the height of the prism is 2cm. Find the volume of the prism.

Answer: 32 \mathrm{cm}^3

1. According to the figure shown in the problem, we can infer that the base of this prism is a rhombus.

2. From the figure in the problem, it can be known that the two diagonals of the rhombus are 8cm and 4cm respectively.

3. Using the theorem: The area of a rhombus can be obtained by its diagonals, i.e., Area = (d1 × d2) / 2, where d1 and d2 are the two diagonals of the rhombus. In this problem, d1 = 8cm, d2 = 4cm.

4. Substituting the diagonals into the formula, we get Area = (8cm × 4cm) / 2 = 16 cm².

5. Using the volume formula: The volume V of the prism = base area × height. In this problem, the area of the rhombus is the base area, and the height of the prism is known to be 2cm.

6. Calculating the volume: V = 16 cm² × 2cm = 32 cm³.

7. Through the above reasoning, the final answer is 32 cm³.

Definition of Rhombus

A quadrilateral is a rhombus if and only if all four sides are of equal length, and its diagonals are perpendicular bisectors of each other.

In the figure of this problem, <note>all sides of the quadrilateral are equal</note>, so the quadrilateral is a rhombus. Additionally, the quadrilateral's <note>diagonals are perpendicular bisectors of each other</note>, meaning that the <note>diagonals intersect at point O</note>, and <note>angles AOB, BOC, COD, and DOA are all right angles (90 degrees)</note>, and <note>diagonals AC and BD bisect each other</note>.

Definition of Prism

A prism is a polyhedron with two parallel and congruent polygonal bases, and all other faces are parallelograms.

In the figure of this problem, <note>the base of the prism is a rhombus</note>, and <note>the height of the prism is 2cm</note>.

Rhombus Area Formula

The area of a rhombus is equal to half the product of its diagonals.

In the figure of this problem, <note>the two diagonals of the rhombus are 8cm and 4cm respectively</note>. According to the rhombus area formula, <note>the area of the rhombus is equal to half the product of the two diagonals</note>, that is, Area = (8cm × 4cm) / 2 = 16 cm².

Volume Formula of Prism

The volume of a prism is equal to the base area multiplied by the height.

<note>The base area of the prism is the area of a rhombus, which is 16 cm²</note>, and <note>the height of the prism is 2 cm</note>. Therefore, the volume of the prism can be calculated using this theorem, <note>V = 16 cm² × 2 cm = 32 cm³</note>.

Definition of Rhombus

<note>In the diagram of this problem, all sides of the quadrilateral are equal, so the quadrilateral is a rhombus</note>. Additionally, <note>the diagonals of the quadrilateral are perpendicular bisectors of each other</note>, meaning <note>the diagonals intersect at point O</note>, and <note>angles AOB, BOC, COD, and DOA are all right angles (90 degrees)</note>, and <note>diagonals AC and BD bisect each other</note>.

<note>In the figure of this problem, all sides of the quadrilateral are equal</note>, therefore, <note>the quadrilateral is a rhombus</note>. Additionally, <note>the diagonals of the quadrilateral bisect each other at right angles</note>, that is, the diagonals <note>intersect at point O</note>, and <note>angles AOB, BOC, COD, and DOA are all right angles (90 degrees)</note>, and <note>the diagonals AC and BD bisect each other</note>.


Question:
Which two triangles are congruent by the ASA Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △ABC≅△SRT

1. Observe the figure and identify the pairs of triangles that need to be verified. The candidate triangles are △ABC and △RST.

2. From the given information, ∠B and ∠R, AB and RS, ∠A and ∠S are congruent respectively, represented as:

   - ∠B = ∠R

   - AB = RS

   - ∠A = ∠S

3. According to the Angle-Side-Angle (ASA) Theorem, if two angles and the included side of one triangle are congruent to the corresponding angles and included side of another triangle, then the two triangles are congruent.

4. In △ABC, ∠B and ∠A are the two angles that include side AB.

5. In △RST, ∠R and ∠S are the two angles that include side RS.

6. Therefore, the two triangles are congruent in terms of two angles and the included side.

7. Match the corresponding vertices:

   - Vertex B corresponds to R

   - Vertex A corresponds to S

   - Therefore, vertex C corresponds to T

8. Based on the Angle-Side-Angle congruence criterion, we conclude △ABC ≅ △SRT.

9. After the above reasoning, the final conclusion is △ABC ≅ △SRT.

<note>∠B=∠R</note>, <note>∠A=∠S</note>, <note>AB=RS</note>. Since two angles <note>∠A and ∠B</note> and the included side <note>AB</note> of △ABC correspond to two angles <note>∠RST and ∠RTS</note> and the included side <note>RS</note> of △RST respectively, according to the Angle-Side-Angle criterion, these two triangles are congruent.


Question: If an angle is bisected to form two new 34° angles, what was the measure of the original angle? $\Box$ °

Answer:
| --- |
| 68° |

1. Given that an angle is bisected into two new angles, each with a measure of 34°.

2. Since this angle is bisected into two equal angles, according to the definition of an angle bisector, the measure of the original angle is equal to the sum of the measures of these two equal angles.

3. Because the measures of these two new angles are both 34°, the measure of the original angle is equal to the sum of 34° and 34°.

4. Calculate the measure of the original angle, 34° + 34° = 68°.

5. Through the above reasoning, the final answer is that the measure of the original angle is 68°.

Definition of Angle Bisector

<note>In the diagram of this problem, an angle is bisected by a line forming two equal angles, each measuring 34°</note>. Therefore, <note>this line is the angle bisector of the angle</note>.


Question: If an angle is bisected to form two new 18° angles, what was the measure of the original angle? $\Box$ °

Answer:
| --- |
| 36° |

1. Given that angle ABC is bisected, forming two angles ABD and DBC, and angle ABD equals 18°, angle DBC also equals 18°.

2. According to the definition of an angle bisector, the angle bisector divides the angle into two equal angles. Therefore, angle ABD equals angle DBC.

3. According to the angle addition principle, angle ABC equals angle ABD plus angle DBC, i.e., m∠ABC = m∠ABD + m∠DBC.

4. Substitute the degrees of angle ABD and angle DBC into the calculation, m∠ABD = 18°, m∠DBC = 18°, thus m∠ABC = 18° + 18°.

5. Therefore, the original angle ABC measures 36°.

6. Through the above reasoning, the final answer is 36°.

<note>The vertex of angle ABC is point B</note>, from <note>point B a line BD is drawn</note>, this line divides <note>angle ABC into two equal angles</note>, namely <note>angle ABD and angle DBC are equal</note>. Therefore, <note>line BD is the angle bisector of angle ABC</note>.


Question:
Look at the diagram.

Which equation can be used to solve for x?

- 5x+65=90
- 5+x=65
- 5x+65=180
- 5x=65
Solve for x.

x= $\Box$

Answer:
- 5x=65
x=13

1. Given ∠AEB = (5x)° and ∠CED = 65°, according to the definition of vertical angles, ∠AEB and ∠CED are vertical angles and their measures are equal.

2. Therefore, in this problem, ∠AEB = ∠CED.

3. Substituting the given conditions, we get (5x)° = 65°.

4. To solve for x, divide both sides of the equation (5x)° = 65° by 5, resulting in x = 13.

5. Through the above reasoning, the final answer is x = 13.

Definition of Vertical Angles

Vertical angles are the pair of opposite angles formed when two lines intersect. These angles are equal in measure.

In the diagram of this problem, <note>two intersecting lines AB and CD intersect at point E</note>, forming four angles: <note>∠AEB, ∠AEC, ∠CED, and ∠BED</note>. According to the definition of vertical angles, <note>∠AEB and ∠CED are vertical angles</note>, <note>∠AEC and ∠BED are vertical angles</note>. Since the angles of vertical angles are equal, <note>∠AEB = ∠CED</note>, <note>∠AEC = ∠BED</note>.


Question: After bisecting an angle, Erica has two new angles, each with a measure of 97°. What was the measure of the original angle? $\Box$ °

Answer:
| --- |
| 194° |

1. Given that Erica bisects an angle and obtains two new angles, each measuring 97°. Assume the original angle is \( \angle ABC \), and after bisecting, it becomes \( \angle ABD \) and \( \angle DBC \).

2. According to the definition of angle bisector, \( \angle ABD = \angle DBC \).

3. The condition given in the problem is that each new angle measures 97°, i.e., \( \angle ABD = 97° \) and \( \angle DBC = 97° \).

4. According to the problem, the measure of the original angle \( \angle ABC \) is: \( \angle ABC = \angle ABD + \angle DBC \).

5. Substituting the values of \( \angle ABD \) and \( \angle DBC \), we get \( \angle ABC = 97° + 97° \).

6. Therefore, \( \angle ABC = 194° \).

7. Through the above reasoning, the final answer is \( \angle ABC = 194° \).

Definition of Angle Bisector

In the figure of this problem, it is known that the original angle is <note> \( \angle ABC \) </note>. After being bisected by the angle bisector <note>BD</note>, two new angles are obtained: <note> \( \angle ABD \) </note> and <note> \( \angle DBC \) </note>. According to the definition of angle bisector, the degrees of <note> \( \angle ABD \) </note> and <note> \( \angle DBC \) </note> are equal. That is, <note> \( \angle ABD = \angle DBC \) </note>.

Definition of an Angle

An angle is a geometric figure formed by two rays sharing a common endpoint. This common endpoint is called the vertex of the angle, and the two rays are referred to as the sides of the angle.

In the diagram of this problem, <note>angle ABC is a geometric figure composed of two rays AB and AC</note>, these two rays have a <note>common endpoint A</note>. This <note>common endpoint A is called the vertex of angle ABC</note>, and <note>rays AB and AC are called the sides of angle ABC</note>.


Question:
Which two triangles are congruent by the ASA Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △WXY≅△CBD

1. In triangles △WXY and △BCD, observe the given conditions: ∠W ≃ ∠C, ∠X ≃ ∠B, and WX ≃ BC.

2. According to the ASA (Angle-Side-Angle) criterion for triangle congruence: If two triangles have two pairs of corresponding angles equal and the side between these angles also equal, then the two triangles are congruent.

3. In △WXY, ∠W and ∠X are known equal angles, and the corresponding side between them is WX.

4. In △BCD, ∠C and ∠B are known equal angles, and the corresponding side between them is BC.

5. Therefore, according to the ASA criterion for triangle congruence, triangle △WXY is congruent to triangle △CBD.

6. To write the congruence relation, align the corresponding vertices: Since ∠W ≃ ∠C and ∠X ≃ ∠B, W corresponds to C, X corresponds to B, so Y corresponds to D. Therefore, the congruence relation is: △WXY ≃ △CBD.

7. Through the above reasoning, the final answer is: △WXY ≃ △CBD.

In the diagram of this problem, <note>∠WXY is a geometric figure formed by rays WX and XY, which share a common endpoint X.</note> This common endpoint X is called the <note>vertex of angle ∠WXY</note>, and <note>rays WX and XY are called the sides of angle ∠WXY</note>. Similarly, <note>∠XWY is a geometric figure formed by rays XW and WY, which share a common endpoint W.</note> This common endpoint W is called the <note>vertex of angle ∠XWY</note>, and <note>rays XW and WY are called the sides of angle ∠XWY</note>.

If two angles and the included side of one triangle are respectively equal to two angles and the included side of another triangle, then the two triangles are congruent.

<note>∠W ≃ ∠C</note>, <note>∠X ≃ ∠B</note>, and <note>side WX ≃ side BC</note>. Since the two triangles have two angles and the included side equal respectively, according to the Angle-Side-Angle (ASA) Criterion for Congruence of Triangles, it can be concluded that <note>triangle WXY is congruent to triangle BCD</note>, thus <note>△WXY ≃ △BCD</note>.

Definition of Congruent Triangles

In the figure of this problem, <note>triangle WXY and triangle CBD are congruent triangles</note>, <note>the corresponding sides and corresponding angles of triangle WXY are equal to those of triangle CBD</note>, namely:
<note>side WX = side BC</note>
<note>side XY = side BD</note>
<note>side WY = side CD</note>
At the same time, the corresponding angles are also equal:
<note>angle WXY = angle CBD</note>
<note>angle XYW = angle BDC</note>
<note>angle YWX = angle DCB</note>.


Question: What is the measurement of this angle? $\Box$ °

Answer: 110°

1. Observe the given figure, let the angle be AOB, the horizontal line be OA, and see that one of the rays aligns with the 0° mark on the inner circle of the protractor, i.e., the ray coincides with the horizontal line of the protractor.

2. The other ray forms an angle with the inner circle marks of the protractor. According to the inner circle marks of the protractor, the extension of this ray is at the 110° position on the inner circle of the protractor.

3. Name this ray as ray OB, and represent the formed angle as angle AOB.

4. Based on the marks at the intersection of ray OA and ray OB on the protractor, the 110° mark on the inner circle of the protractor determines the angle measurement value of angle AOB to be 110°.

5. Therefore, the size of angle AOB is 110° as read from the inner circle marks of the protractor.

7. After the above reasoning, the final answer is <110°>.

Measurement of Angle

Align the baseline of the protractor with one side of the angle, then extend the other side to the protractor's scale. Determine the angle by reading the difference between the graduations of the two sides.

In the diagram of this problem, let <note>the vertex of angle AOB be point O</note>, <note>the two sides of the angle be segment AO and segment BO respectively</note>. When measuring <note>angle AOB</note> using a protractor, align <note>the center point of the protractor with the vertex of angle AOB</note>, align <note>the inner scale line of the protractor with segment AO so that it coincides with the inner scale line of the protractor</note>. Then, <note>extend segment BO to the scale plate of the protractor, and determine the degree of angle AOB by reading the difference in scales between segment AO and segment BO</note>.


Question:
The figure above shows a triangular prism with height $\sqrt{3} x$. If the volume of the prism is $\frac{81}{4}$, what is the value of $x$ ?
Choices:
A:3
B:4
C:5
D:6

Answer: A

1. First, we understand that the volume formula of a prism is the base area multiplied by the height, i.e., Volume = Base Area × Height.

2. It is known that one base of the triangular prism is an equilateral triangle with side length x. Since the area formula of an equilateral triangle is (√3 / 4) × side length^2, the base area = (√3 / 4) × x^2.

3. In the problem, the height of the prism is √3 × x.

4. Substitute the provided information into the volume formula, the volume is V = ((√3 / 4) × x^2) × (√3 × x).

5. Simplify to V = (3√3 / 4) × x^3.

6. According to the specific value of the volume given in the problem, V is equal to 81/4.

7. Set up the equation (3√3 / 4) × x^3 = 81/4.

8. Multiply both sides by 4/3√3 to get x^3 = 27.

9. By solving the equation, we find that x cubed equals 27, i.e., x = 3.

10. According to the options, x = 3 corresponds to option A.

11. Based on the above reasoning, the final answer is x = 3.

Definition of Equilateral Triangle

An equilateral triangle is a triangle in which all three sides have the same length, and all interior angles are equal (each measuring 60°).

<note>The base triangle is an equilateral triangle</note>. The side lengths are <note>x, x, and x</note>, <note>each interior angle is 60°</note>.

Area Formula of a Triangle

The area of any triangle is equal to its base multiplied by its height, divided by 2.

<note>The side length of the equilateral triangle base is x</note>, according to the area formula of an equilateral triangle, <note>base area = (√3 / 4) × x^2</note>.

Volume Formula of Prism

<note>The base area of the triangular prism is (√3 / 4) × x^2</note>, the height is <note>√3 x</note>, so the <note>volume of the triangular prism is (√3 / 4) × x^2 × √3 x</note>.


Question:
Look at the diagram.

Which equation can be used to solve for x?

- 5+x=65
- 5x+65=90
- 5x+65=180
- 5x=65
Solve for x.

x= $\Box$

Answer:
- 5x=65
x=13

1. According to the problem statement and the markings in the figure, we notice that point E is the intersection of two crossed lines, thus forming two vertical angles.

2. According to the definition of vertical angles, when two lines intersect, the two opposite angles formed are vertical angles, and their degrees are equal.

3. From the figure, we know that ∠AEB = (5x)°, ∠CED = 65°, and these two angles are vertical angles, so we have (5x)° = 65°.

4. From the equation (5x)° = 65°, dividing both sides by 5, we calculate x = 13.

5. Through the above reasoning, we finally obtain the answer x = 13.

Definition of Vertical Angles

In the figure of this problem, <note>two intersecting lines AD and BC intersect at point E, forming four angles: ∠AEB, ∠CED, ∠BED, and ∠AEC</note>. According to the definition of vertical angles, <note>∠AEB and ∠CED are vertical angles</note>, <note>∠BED and ∠AEC are vertical angles</note>. Since the angles of vertical angles are equal, <note>∠AEB=∠CED</note>, <note>∠BED=∠AEC</note>. Therefore, <note>(5x)° = 65°</note>.


Question:
Look at the diagram.

Which equation can be used to solve for x?

- 4x=76
- 4x+76=180
- 4+x=76
- 4x+76=90
Solve for x.

x= $\Box$

Answer:
- 4x=76
x=19

1. In the figure, we see that line DG and line EF intersect at point H.

2. According to the definition of vertical angles, ∠DHF and ∠EHG have equal measures, i.e., ∠DHF = ∠EHG.

3. It is known that the measure of ∠DHF is 76°, and ∠EHG is represented as 4x. Based on the above points, we can obtain the equation: 4x = 76.

4. Solve the equation 4x = 76:

5. Divide both sides by 4 to get x = 76/4.

6. Calculate to get x = 19.

7. Through the above reasoning, the final answer is x = 19.

<note>Two intersecting lines DH and EH intersect at point H</note>, forming four angles: <note>∠DHF, ∠EHG, ∠DHE, and ∠GHF</note>. According to the definition of vertical angles, <note>∠DHF and ∠EHG are vertical angles</note>, <note>∠DHE and ∠GHF are vertical angles</note>. Since vertical angles are equal, <note>∠DHF = ∠EHG</note>, <note>∠DHE = ∠GHF</note>. Given that <note>∠DHF = 76 degrees</note>, therefore <note>∠EHG = 4x</note>, then <note>4x = 76</note>.


Question:
Look at the diagram.

Which equation can be used to solve for x?

- 5x+65=180
- 5+x=65
- 5x=65
- 5x+65=90
Solve for x.

x= $\Box$

Answer:
- 5x=65
x=13

1. Given that the lines in the figure intersect at point E, forming two vertical angles, namely angle AEB and angle CED.

2. According to the definition of vertical angles, the angles AEB and CED are equal. Therefore, we can obtain the equation: ∠AEB = ∠CED. At this point, ∠AEB = (5x)°, ∠CED = 65°.

3. Based on the definition of vertical angles, in this problem it specifically means (5x)° = 65°.

4. Solve the equation 5x = 65, to get x = 65/5.

5. Calculate to get x = 13.

6. Through the above reasoning, the final answer is x = 13.

In the figure of this problem, <note>line AD and line BC intersect at point E</note>, forming four angles: <note>∠AEB</note>, <note>∠CED</note>, ∠BED, and ∠AEC. According to the definition of vertical angles, <note>∠AEB and ∠CED are vertical angles</note>, and ∠BED and ∠AEC are vertical angles. Since the angles of vertical angles are equal, <note>∠AEB = ∠CED</note>, that is, <note>(5x)° = 65°</note>.


Question: What is the measurement of this angle? $\Box$ °

Answer: 85°

1. According to the protractor image in the problem, confirm that the base edge of the protractor (horizontal) is aligned with the 0° mark.

2. Find the two rays forming the angle, one ray extends along the 0° mark, and the other ray points to a certain mark on the protractor.

3. Carefully observe the protractor and find the point where the second ray intersects with the upper semicircle scale.

4. Pay close attention to the reading of the scale on the protractor and confirm the intersection point of the second ray with the 85° mark, indicating that the measured angle is 85°.

5. Based on the above observations and measurements, it can be determined that the size of the angle is 85°.

6. Through the above reasoning, the final answer is 85°.

Measurement of Angle

In the figure of this problem, <note>the angle is formed by rays OA and OB</note>, <note>with endpoint O being the common endpoint</note>. Ray OA follows the <note>0° mark</note> of the protractor, and ray OB points to the <note>85° mark</note>, forming an <note>85° angle</note>.


Question: What is the measurement of this angle? $\Box$ °

Answer: 120°

1. Find the ray that coincides with 0° on the protractor, which is one of the rays, labeled as OX.

2. Find the other ray, which extends from the center of the protractor in the other direction, labeled as OY.

3. Using the outer degree scale of the protractor, read the measurement from OX to OY in a clockwise direction, it can be seen that the ray OY points to 120°.

4. According to the measurement of the angle, the size of angle ∠XOY is the degree from 0° to 120°.

5. Therefore, we can directly read the degree of this angle as 120°.

6. Through the above reasoning, the final answer is 120°.

Definition of an Angle

<note>The angle ∠XOY is a geometric figure formed by ray OX and ray OY</note>, <note>these two rays share a common endpoint O</note>. <note>This common endpoint O is called the vertex of angle ∠XOY</note>, and <note>ray OX and ray OY are called the sides of angle ∠XOY</note>.

Measurement of Angle

<note>Align the center point of the protractor with vertex O</note>, ray OX aligns with the <note>0° mark</note> on the protractor, ray OY points to the <note>120° mark</note> on the protractor. Therefore, the angle <note>∠XOY is 120°</note>.


Question: If an angle is bisected to form two new 82° angles, what was the measure of the original angle? $\Box$ °

Answer:
| --- |
| 164° |

1. Given that angle \( \angle ABC \) is bisected, forming two angles: \( \angle ABD \) and \( \angle DBC \), and \( \angle ABD = \angle DBC = 82^\circ \).

2. According to the definition of angle bisector, the angle bisector divides an angle into two equal angles, thus \( \angle ABC = \angle ABD + \angle DBC \).

3. Substituting the given conditions, since \( \angle ABD = 82^\circ \) and \( \angle DBC = 82^\circ \), we get \( \angle ABC = 82^\circ + 82^\circ \).

4. Calculating, we get \( \angle ABC = 164^\circ \).

5. Therefore, through the above reasoning, the final answer is \( 164^\circ \).

Definition of an Angle

In the diagram of this problem, <note>angle \( \angle ABC \)</note> is a geometric figure formed by two <note>rays \(AB\) and \(BC\)</note>, which share a <note>common endpoint \(B\)</note>. This <note>common endpoint \(B\)</note> is called the <note>vertex of angle \( \angle ABC \)</note>, and <note>rays \(AB\) and \(BC\)</note> are called the <note>sides of angle \( \angle ABC \)</note>.

Definition of Angle Bisector

An angle bisector is a line that originates from the vertex of an angle and divides the angle into two congruent angles.

In the figure of this problem, <note>the vertex of angle \( \angle ABC \) is point B</note>, a line BD is drawn from <note>point B</note>, <note>this line divides angle \( \angle ABC \) into two equal angles</note>, that is, <note>\( \angle ABD \) and \( \angle DBC \) are equal</note>. Therefore, <note>line BD is the angle bisector of angle \( \angle ABC \)</note>.


Question:
Which two triangles are congruent by the AAS Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △FGH≅△ZXY

1. Given that \( \angle G \cong \angle X \), this is the first pair of equal angles.

2. Given that \( \angle F \cong \angle Z \), this is the second pair of equal angles.

3. Given that segment \( \overline{GH} \cong \overline{XY} \), this is the side not included between these two pairs of angles.

4. According to the AAS (Angle-Angle-Side) congruence theorem, if two pairs of angles and a side not included between these angles are equal in two triangles, then these two triangles are congruent.

5. In this problem, triangles \( \triangle FGH \) and \( \triangle XYZ \) respectively satisfy the above conditions, hence they are congruent.

6. To clarify the corresponding vertices of the congruent triangles, given that \( \angle G \cong \angle X \) and \( \angle F \cong \angle Z \), G corresponds to X, F corresponds to Z, hence H corresponds to Y.

7. Based on the above analysis and reasoning, the congruence relationship of the triangles can be written as \( \triangle FGH \cong \triangle ZXY \).

8. After the above reasoning, the final answer is \( \triangle FGH \cong \triangle ZXY \).

Definition of an Angle

<note>Angle G is a geometric figure composed of two rays HG and FG</note>, <note>these two rays share a common endpoint G</note>. This <note>common endpoint G is called the vertex of angle G</note>, and <note>rays HG and FG are called the sides of angle G</note>. <note>Angle F is a geometric figure composed of two rays HF and FG</note>, <note>these two rays share a common endpoint F</note>. This <note>common endpoint F is called the vertex of angle F</note>, and <note>rays HF and FG are called the sides of angle F</note>. <note>Angle X is a geometric figure composed of two rays XZ and XY</note>, <note>these two rays share a common endpoint X</note>. This <note>common endpoint X is called the vertex of angle X</note>, and <note>rays XZ and XY are called the sides of angle X</note>. <note>Angle Z is a geometric figure composed of two rays XZ and ZY</note>, <note>these two rays share a common endpoint Z</note>. This <note>common endpoint Z is called

Definition of Congruent Triangles

In the figure of this problem, triangle \( \triangle FGH \) and triangle \( \triangle ZXY \) are congruent triangles, <note>the corresponding sides and corresponding angles of triangle \( \triangle FGH \) are equal to those of triangle \( \triangle ZXY \)</note>, that is: <note>\( \overline{FG} = \overline{ZX} \)</note>, <note>\( \overline{GH} = \overline{XY} \)</note>, <note>\( \overline{FH} = \overline{ZY} \)</note>, and at the same time, <note>the corresponding angles are also equal</note>: <note>\( \angle FGH = \angle ZXY \)</note>, <note>\( \angle GHF = \angle XYZ \)</note>, <note>\( \angle HFG = \angle YZX \)</note>.

Congruence Theorem for Triangles (AAS)

Two triangles are congruent if two angles and the included side of one triangle are respectively equal to two angles and the included side of the other triangle.

<note>Angle G is equal to angle Z</note>, <note>angle F is equal to angle Z</note>, <note>side GF is equal to side XZ, and this side is not between the two known angles</note>. Therefore, according to the Congruence Theorem for Triangles (AAS), <note>triangle FGH is congruent to triangle ZXY</note>.


Question:
Look at this angle:

What is the vertex?
 $\Box$

Answer: T

1. In geometry, an angle is a figure formed by two rays sharing a common starting point. This common starting point is called the vertex of the angle.

2. Observing the angle given in the figure, it is composed of two rays ST and TU, where S, T, and U are points on these rays.

3. In rays ST and TU, the common starting point is point T.

4. According to the definition of an angle, point T is the common starting point of these two rays, therefore point T is the vertex of this angle.

5. Through the above reasoning, the final answer is T.

Definition of an Angle

<note>Angle ∠STU is a geometric figure composed of two rays ST and TU</note>, these two rays share a common endpoint <note>T</note>. This common endpoint <note>T is called the vertex of angle ∠STU</note>, and the rays <note>ST and TU are called the sides of angle ∠STU</note>.


Question: What is the measurement of this angle? $\Box$ °

Answer: 155°

1. Observe the given image and find that there is a semicircular protractor used to measure the size of the angle.

2. According to the measurement of the angle, align the center point of the protractor with the vertex of the angle, and make the horizontal base of the protractor coincide with one side of the angle, starting from the 0° mark.

3. From the figure, the degrees corresponding to the base of the protractor increase from 0° to 180° from left to right.

4. For the other side starting from 0°, read the position of the intersection point corresponding to the scale on the base of the protractor, and it can be intuitively seen that the scale pointed by this side is 155°.

5. Since the protractor does not require reverse or inverted reading, the measured degree is read in the forward direction.

6. Knowing that the forward direction starts from 0° until the protractor intersects with the other side of the angle, the reading is 155°, so the angle is 155°.

7. After the above reasoning, the final answer is 155°.

<note>An angle is a geometric figure formed by two rays</note>, these two rays <note>share a common endpoint (the center point of the figure)</note>. This common endpoint is called <note>the vertex of the angle</note>, and <note>the rays are respectively the horizontal black line starting from the 0° mark and another ray extending to the 155° mark</note>, they form the angle we need to measure.

Measurement of Angle

In this problem, apply the angle measurement principle, <note>align the center point of the protractor with the vertex of the angle</note>, <note>align the baseline of the protractor with one side of the angle (horizontal black line) at the 0° mark</note>, read the position where the other side of the protractor (diagonal line) intersects the scale, which shows <note>155°</note>, so the measure of the angle is <note>155°</note>.


Question: What is the measurement of this angle? $\Box$ °

Answer: 75°

1. Use a protractor to measure the given angle. In the figure, one ray aligns with the 0° line of the inner circle of the protractor, which corresponds to 180° on the outer circle.

2. Observe the position where the other ray intersects with the inner circle of the protractor.

3. Read the position of the angle from the inner circle of the protractor, its scale shows 75°.

4. From the above steps, it is concluded that the measured angle is 75°.

Definition of an Angle

<note>An angle is a geometric figure formed by two rays</note>, these two rays have a common endpoint, which is <note>the center point of the protractor</note>. This common endpoint is called <note>the vertex of the angle</note>, and the rays are respectively <note>the ray aligned with the 0° line of the inner circle of the protractor</note> and <note>the ray intersecting the 75° line of the inner circle of the protractor</note>, called <note>the sides of the angle</note>.

Measurement of Angle

<note>In the diagram of this problem</note>, we align <note>one ray with the 0° line of the inner circle of the protractor</note>, then observe <note>the position where the other ray intersects the 75° line of the inner circle of the protractor</note>, thus obtaining <note>an angle of 75°</note>.


Question: What is the measurement of this angle? $\Box$ °

Answer: 130°

1. First, observe the protractor's scale and find that the inner ring scale increases from left to right, while the outer ring scale increases from right to left.

2. In the figure, the right ray aligns with the 0° diameter on the inner ring. Then, observe the scale line that the left ray passes through and find that this ray passes through 130° on the inner ring.

3. Therefore, the angle can be directly read as 130° on the inner ring.

4. Since the position of the rays ensures that the measurement is taken on the inner ring, there is no need to correct the angle.

5. Through the above analysis, it can be confirmed that the angle in the figure is 130°.

6. According to the rules of using a protractor, the measured angle directly reflects the angle between the two rays.

7. After the above reasoning, the final answer is 130°.

Definition of an Angle

In the diagram of this problem, an angle is a geometric figure formed by two rays, which are <note>from the center of the protractor pointing to the inner ring 0°</note> and <note>the ray at the inner ring 130°</note>. These two rays share a common endpoint, called <note>the vertex of the angle, located at the center of the protractor</note>. <note>The rays from the center of the protractor pointing to the inner ring 0° and the inner ring 130° are called the sides of the angle</note>.

Measurement of Angle

Align the <note>center point of the protractor with the vertex of the angle</note>, make <note>one of the rays align with the 0° mark on the inner ring</note>, the other ray passes through <note>130° on the inner ring</note>, thus the measured angle is <note>130°</note>.