Question: Using parallelogram ABCD, find m \angle DAC.

Answer: 72

1. In triangle ADF, according to the triangle angle sum theorem, it is known that ∠ADF + ∠AFD + ∠DAF = 180°.

2. Substitute the known values ∠ADF = 59°, ∠AFD = ∠BFC = 49° (according to the definition of vertical angles), and calculate ∠DAC = 180° - ∠ADF - ∠AFD = 180° - 59° - 49° = 72°.

3. Therefore, the value of ∠DAC is 72°.

平行四边形; 三角形的外角; 等腰三角形

Triangle Angle Sum Theorem

<note>In triangle DAF, angle DAF, angle AFD, and angle ADF</note> are the three interior angles of triangle DAF. According to the Triangle Angle Sum Theorem, <note>angle DAF + angle AFD + angle ADF = 180°</note>.

Definition of Vertical Angles

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

<note>In the diagram of this problem, two intersecting lines AC and ##BD## intersect at point F</note>, forming four angles: <note>##∠AFD, ∠AFB##, ∠DFC, and ∠BFC</note>. According to the definition of vertical angles, <note>##∠AFD## and ∠BFC are vertical angles</note>, <note>##∠AFB## and ∠DFC are vertical angles</note>. Since the angles of vertical angles are equal, <note>##∠AFD## = ∠BFC</note>, <note>##∠AFB## = ∠DFC</note>.


Question: In the figure above, ABCD is a rectangle. What is the area of ABCD?

Answer: \sqrt{3}

1. Given conditions: ABCD is a rectangle, and the length of diagonal AC is 2. According to the definition of a rectangle, the interior angles ∠BAD=90°, ∠ABC=90°. Given ∠DAC=30°, so ∠BAC=∠BAD-∠DAC=90°-30°=60°, thus ∠BAC is 60°. Since ∠ABC=90°, according to the definition of a right triangle, triangle ABC is a right triangle.

2. According to the definition of the sine function, in right triangle ABC, sin(∠BAC) = opposite side BC / AC, so sin(60°) = BC / 2, that is √3/2 = BC / 2, therefore BC = √3.

3. In right triangle ABC, according to the cosine function, cos(∠BAC) = adjacent side AB / AC, so cos(60°) = AB / 2, that is 1/2 = AB / 2, therefore AB = 1.

4. Given that ABCD is a rectangle, sides AB and BC are adjacent sides. In a rectangle, the product of two adjacent sides is the area of the rectangle, i.e., area S = AB × BC.

5. Substituting the known values, AB = 1, BC = √3, thus S = √3 × 1 = √3.

6. Through the above reasoning, the final answer is √3.

矩形; 直角三角形; 余弦

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 diagram of this problem, <note>quadrilateral ABCD is a rectangle</note>, its <note>interior angles ∠DAB, ∠ABC, ∠BCD, ∠CDA are all right angles (90 degrees)</note>, and <note>sides AB and CD are parallel and equal in length</note>, <note>sides BC and DA are parallel and equal in length</note>.

Definition of Sine Function

In a right-angled triangle, the sine of an acute angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

In the diagram of this problem, in <note>right triangle ABC</note>, <note>angle ∠BAC is an acute angle</note>, <note>side BC is the opposite side of angle ∠BAC</note>, and <note>side AC is the hypotenuse</note>. According to the definition of the sine function, the <note>sine value of angle ∠BAC</note> is equal to the <note>ratio of the opposite side BC to the hypotenuse AC</note>, that is, <note>sin(∠BAC) = BC / AC</note>.

Cosine Function

The cosine function is a trigonometric function that can be defined via the sides of a right triangle. For an angle α, its cosine value is equal to the ratio of the length of the adjacent side to the hypotenuse. In the unit circle, the cosine value is the x-coordinate of the point on the circle that makes an angle α with the positive x-axis.

In the diagram of this problem, in the right triangle ABC, <note>side AB is the adjacent side of angle ∠BAC</note>, <note>side AC is the hypotenuse</note>. Therefore, according to the definition of the cosine function, <note>the cosine value of angle ∠BAC is equal to the ratio of the adjacent side AB to the hypotenuse AC</note>, that is, <note>cos(∠BAC) = AB / AC</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>rectangle ABCD</note>, <note>sides AB and BC</note> are the length and width of the rectangle, so <note>the area of the rectangle = AB * BC</note>.

Definition of Right Triangle

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

<note>In triangle ABC, angle ABC is a right angle (90 degrees)</note>, therefore triangle ABC is a right triangle. <note>Side AB and side BC are the legs</note>, <note>side AC is the hypotenuse</note>.


Question: As shown in the figure, the distance between street lamps 甲 and 乙 is 30 meters. One night, when Xiaogang walked straight from the base of street lamp 甲 to the base of street lamp 乙 for 25 meters, he found that the top of his shadow just touched the base of street lamp 乙. Given that Xiaogang's height is 1.5 meters, the height of street lamp 甲 is ()

Answer: 9米

1. It is known that Xiaogang's height is 1.5 meters. Let the height of Lamp A be AB, and use the symbol CD to represent Xiaogang's height.

2. Connect the top of Lamp A and the top of Xiaogang's shadow to form a straight line. Let the bottom of Lamp B be O, the distance from Xiaogang to Lamp B be 25 meters, the distance between Lamp A and Lamp B be 30 meters, ##the distance between Xiaogang and Lamp B be 5 meters.##

3. Since the distance from Xiaogang to the bottom of Lamp B is a straight line, both Lamp A and Lamp B are perpendicular to the ground, i.e., AB is perpendicular to OB, and CD is perpendicular to OB.

4. ##According to the similarity theorem of triangles (AA), △ABO is similar to △CDO (if two angles in two triangles are equal respectively, then the two triangles are similar).##

5. ##From the above steps, it is concluded that triangles ABO and CDO are similar. The height of AB can be calculated using the similarity theorem of triangles (SAS): AB/CD = BO/DO.##

6. Substitute the known conditions, ##AB/1.5 = 30/5 = 6 = AB/1.5 = 6*1.5 = AB = 9.##

####

##7##. Through the above reasoning, the final answer is: the height of Lamp A is 9 meters.

Similarity Theorem for Triangles (AA)

Two triangles are similar if two of their corresponding angles are congruent.

<note>Angle ABO is equal to angle CDO</note>, and <note>angle AOB is equal to angle COD</note>, so <note>triangle ABO is similar to triangle CDO</note>.

SAS Criterion for Similar Triangles

If two triangles have two pairs of corresponding sides in proportion and the included angle between those sides is equal, then the two triangles are similar.

<note>In the figure of this problem, in triangles ABO and CDO, side AB corresponds to side CD, side BO corresponds to side DO, and side AB/side CD = side BO/side DO, and angle ABO = angle CDO</note>, so <note>according to the Side-Angle-Side (SAS) Criterion for Similar Triangles, triangle ABO is similar to triangle CDO</note>.


Question: Find the area of the figure. Round to the nearest tenth if necessary.

Answer: 35.7

1. The given conditions are that the length of the triangle's ##base## is 10.2 ft, and the ##height## is 7 ft. According to the triangle area formula, the area of the triangle is equal to the base multiplied by the height divided by 2.

2. Using the formula based on the given conditions, let the base be b=10.2 ft and the height be h=7 ft, then the area A=(b * h) / 2.

3. Substitute the specific values: A = (10.2 * 7) / 2.

4. Perform the calculation, 10.2 * 7 equals 71.4, and 71.4 divided by 2 equals 35.7.

5. Therefore, the area of the triangle is 35.7 square feet.

6. As required by the problem, if necessary, round the result to the nearest tenth.

7. The final result is 35.7 square feet, no further rounding is needed.

普通三角形; 直角三角形

Definition of Triangle

A geometric figure is called a triangle if it is formed by three non-collinear points and the line segments connecting them.

<note>A triangle is a geometric figure composed of three non-collinear points and the line segments connecting them</note>. The base of the triangle <note>is 10.2 feet long</note>, <note>and the height is 7 feet</note>.

Definition of Altitude

An altitude is a perpendicular line segment extending from a vertex of a triangle to the opposite side (or to the extension of the opposite side).

In the figure of this problem, <note>the altitude is perpendicular from the vertex of the triangle to the base at 10.2 feet</note>, and its length is 7 feet. <note>The line segment forms a right angle (90 degrees) with the base</note>, which indicates that this line segment is the perpendicular distance from the vertex to the base.

Area Formula of a Triangle

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

<note>The base of the triangle is 10.2 feet</note>, <note>the height is 7 feet</note>. According to the area formula of a triangle, the area of the triangle is equal to <note>the base multiplied by the height and then divided by 2</note>, i.e., Area = (10.2 * 7) / 2.


Question: As shown in the figure, it is known that CD is the diameter of circle O, and the chord DE passing through point D is parallel to the radius OA. If angle D = 50°, then the degree of angle C is ()

Answer: 25°

1. Given that CD is the diameter of circle O, DE is a chord passing through point D and parallel to radius OA, ∠D=50°.

2. Because DE is parallel to OA, ##according to the parallel axiom 2 of parallel lines and the definition of alternate interior angles##, we get ∠AOD=∠D=50°.

3. According to the inscribed angle theorem, the inscribed angle is equal to half of the central angle it subtends, that is, ∠ACD=1/2∠AOD.

4. Substituting the known value of ∠AOD, we get ∠ACD=1/2×50°=25°.

5. Therefore, the measure of angle C is 25°.

圆; 圆周角; 圆心角; 平行线; 弦

Definition of Parallel Lines

Two lines in the same plane that do not intersect are called parallel lines.

In the figure of this problem, <note>line AO and line DE lie in the same plane</note>, and <note>they do not intersect</note>, so according to the definition of parallel lines, <note>line AO and line DE are parallel lines</note>.

Definition of Alternate Interior Angles

When a straight line (referred to as a transversal) intersects two parallel lines, the angles that are located between the two parallel lines but on opposite sides of the transversal are called alternate interior angles.

<note>In the figure of this problem, two parallel lines AO and DE</note> are <note>intersected by a straight line OD</note>, where <note>angle AOD and angle ODE are located between the two parallel lines</note> and on opposite sides of the transversal OD. Therefore, <note>angle AOD and angle ODE are alternate interior angles</note>. Alternate interior angles are equal, i.e., <note>angle AOE is equal to angle ODE</note>.

Definition of Inscribed Angle

An inscribed angle is an angle whose vertex lies on the circumference of a circle and whose sides intersect the circle at two points.

In <note>circle O</note>, <note>the vertex of angle ACD (point C) is on the circumference</note>, <note>the two sides of angle ACD intersect circle O at points A and D respectively</note>. Therefore, <note>angle ACD is an inscribed angle</note>.

Definition of Central Angle

An angle formed by two radii connecting two points on the circumference of a circle to the center of the circle is called a central angle.

In <note>circle O</note>, <note>point A</note> and <note>point D</note> are two points on the circle, and the <note>center of the circle is point O</note>. The angle ∠AOD formed by the lines <note>OA and OD</note> is called the central angle.

Parallel Postulate 2 of Parallel Lines

If two parallel lines are cut by a transversal, then the corresponding angles are equal, the alternate interior angles are equal, and the consecutive interior angles on the same side of the transversal are supplementary.

<note>DE is parallel to OA</note>, therefore according to Parallel Postulate 2 of Parallel Lines, <note>DE and OA cut by line OD form alternate interior angles ∠AOD and ∠ODE that are equal</note>. Since <note>∠ODE=50°</note>, therefore <note>∠AOD=50°</note>.

Inscribed Angle Theorem

In a circle, an inscribed angle is equal to half of the central angle that subtends the same arc.

The central angle corresponding to <note>arc AD</note> is <note>∠AOD</note>, and the inscribed angle is <note>∠ACD</note>. According to the Inscribed Angle Theorem, <note>∠ACD</note> is equal to half of the central angle <note>∠AOD</note> corresponding to <note>arc AD</note>, that is, <note>∠ACD = 1/2 ∠AOD</note>.


Question: As shown in the figure: Two vertical poles AB with a length of 6 and CD with a length of 3 intersect at AD and BC at point E. What is the distance from point E to the ground, EF?

Answer: 2

1. Given AB and CD are two vertical poles, AB has a length of 6, CD has a length of 3, and EF is perpendicular to the ground.

2. According to the problem statement and the diagram, triangles ABD and EFD are similar (##given ∠EFD = ∠ABD, ∠EDF = ∠ADB, using the similarity theorem (AA)##).

3. Similarly, due to the perpendicular relationship, triangles BCD and BEF are also similar (##given ∠EBF = ∠CBD, ∠EFB = ∠CDB, using the similarity theorem (AA)##).

4. According to the ##definition of similar triangles##, corresponding sides are proportional, we get: DF/BD = EF/AB, thus DF/BD = EF/6.

5. Similarly, using the ##definition of similar triangles##, we get: BF/BD = EF/CD, thus BF/BD = EF/3.

6. Adding the two equations, we get: DF/BD + BF/BD = EF/6 + EF/3.

7. Since DF and BF are parts connected to BD, we have DF/BD + BF/BD = 1.

9. Simplifying the equation, we get (1/6 + 1/3) * EF = 1, which means 1/2 * EF = 1.

11. Through the above reasoning, we finally conclude that the length of EF, the distance from E to the ground, is 2.

Definition of Perpendicular Lines

Two lines are said to be perpendicular if and only if the angle formed by their intersection is 90 degrees.

In the figure of this problem, <note>line EF and line BD</note> intersect to form an angle <note>∠EFD is 90 degrees</note>, so according to the <note>Definition of Perpendicular Lines</note>, line EF and line BD are perpendicular to each other.

Definition of Similar Triangles

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

<note>Triangles ABD and EFD are similar triangles</note>. According to the definition of similar triangles: <note>∠BAD = ∠FED</note>, <note>∠ABD = ∠EFD</note>, <note>∠EDF = ∠ADB</note>; <note>AB/EF = BD/FD = AD/ED</note>. Similarly, <note>Triangles BCD and BEF are similar triangles</note>. According to the definition of similar triangles: <note>∠BCD = ∠BEF</note>, <note>∠EBF = ∠CBD</note>, <note>∠EFB = ∠CDB</note>; <note>BC/BE = CD/EF = BD/BF</note>.

Similarity Theorem for Triangles (AA)

<note>In the figure of this problem, in triangles ABD and EFD, ∠EDF = ∠ADB, and ∠ABD = ∠EFD, so triangle ABD is similar to triangle EFD</note>. <note>Similarly, in triangles BCD and BEF, ∠EBF = ∠CBD, and ∠BDC = ∠BFE, so triangle BCD is similar to triangle BEF</note>.


Question: Find z.

Answer: 12

1. Given that △RST is an isosceles triangle with side lengths ST = 7, RT = 9, SR = 2z - 15. We need to find the value of z.

2. In the isosceles triangle RST, ∠S = ∠T, RT is the side opposite ∠S, and RS is the side opposite ∠T. According to the properties of isosceles triangles, SR = RT.

3. Substitute SR = 2z - 15 and RT = 9 into the equation, we get 2z - 15 = 9.

4. Solve the equation 2z - 15 = 9, we get z = 12.

5. Through the above reasoning, the final answer is z = 12.

Properties of Isosceles Triangle

In an isosceles triangle, the angles opposite the equal sides are equal, and the sides opposite the equal angles are congruent.

In the <note>isosceles triangle RST</note>, <note>angle S and angle T are equal</note>. Therefore, according to the <note>properties of isosceles triangles</note>, <note>the sides opposite the equal angles are equal, i.e., side RS = side RT</note>.

Definition of Isosceles Triangle

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

<note>Side RS and side RT are equal</note>, so <note>triangle RST is an isosceles triangle</note>.


Question: Find y.

Answer: 6 \sqrt { 3 }

1. Let the three vertices of the triangle in the figure be A, B, and C. From the figure, we get: ∠ABC=90°, ∠BCA=30°, AB=6.

2. Given ∠ABC=90°, ∠BCA=30°, according to the triangle angle sum theorem, ∠ABC+∠BCA+∠CAB=90°+30°+∠CAB=180°, i.e., ∠CAB=180°-90°-30°=60°.

3. Also given AB=6, BC=y, according to the properties of a 30°-60°-90° triangle, in a 30°-60°-90° triangle, the side opposite the 60° angle is √3 times the shorter side, i.e., BC=6√3, thus y=6√3.

5. After the above reasoning, the final answer is 6√3.

直角三角形; 正弦; 余弦

Definition of Right Triangle

<note>In the figure of this problem, in triangle ABC, angle ABC is a right angle (90 degrees)</note>, therefore triangle ABC is a right triangle. <note>Side AB and side BC are the legs, side AC is the hypotenuse</note>.

Properties of 30°-60°-90° Triangle

In a 30°-60°-90° triangle, the side opposite the 30-degree angle (the shorter leg) is half the length of the hypotenuse. The side opposite the 60-degree angle (the longer leg) is √3 times the length of the shorter leg.

<note>Angle BCA is 30 degrees</note>, <note>Angle CAB is 60 degrees</note>, <note>Angle ABC is 90 degrees</note>. <note>Side AC is the hypotenuse</note>, <note>Side AB is the side opposite the 30-degree angle</note>, <note>Side BC is the side opposite the 60-degree angle</note>. According to the properties of a 30°-60°-90° triangle, <note>Side AB is half of Side AC</note>, <note>Side BC is √3 times Side AB</note>. That is: <note>AC = 2 * AB</note>, <note>BC = AB * √3</note>.

Triangle Angle Sum Theorem

The sum of the interior angles of any triangle is 180°.

In the <note>triangle</note>, <note>angle 30°</note>, <note>angle 90°</note>, and <note>angle 60°</note> are the three interior angles of the triangle. According to the Triangle Angle Sum Theorem, <note>30° + 60° + 90° = 180°</note>.


Question: As shown in the figure, in △ABC, AB=AC, and AD∥BC is drawn through point A. If ∠1=70°, then the measure of ∠BAC is ()

Answer: 40°

1. According to the problem, AD∥BC. Based on the parallel axiom 2 of parallel lines, alternate interior angles are equal, thus ∠1 = ∠ACB = 70°.

2. In △ABC, it is known that AB=AC. According to the definition of an isosceles triangle, △ABC is an isosceles triangle. Based on the properties of an isosceles triangle, we get ∠ABC=∠ACB=70°.

3. According to the triangle angle sum theorem: the sum of the three interior angles of a triangle is equal to 180°, i.e., ∠BAC + ∠ABC + ∠ACB = 180°.

4. Substituting the known conditions, we get ∠BAC + 70° + 70° = 180°.

5. Therefore, we can find ∠BAC = 180° - 70° - 70° = 40°.

6. Through the above reasoning, the final answer is 40°.

等腰三角形; 平行线; 内错角

Definition of Isosceles Triangle

In the figure of this problem, in triangle ABC, <note>sides AB and AC are equal</note>, therefore the triangle <note>ABC is an isosceles triangle</note>.

Parallel Postulate 2 of Parallel Lines

<note>In the diagram of this problem, two parallel lines AD and BC are intersected by a third line AC</note>, forming the following geometric relationship: <note>Alternate interior angles: angle 1 and angle ACB are equal</note>.

Properties of Isosceles Triangle

<note>In the figure of this problem, in the isosceles triangle ABC, side AB and side AC are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, <note>i.e., angle ABC = angle ACB</note>.

Triangle Angle Sum Theorem

In the diagram of this problem, in triangle ABC, <note>angle BAC</note>, <note>angle ABC</note>, and <note>angle ACB</note> are the three interior angles of triangle ABC. According to the Triangle Angle Sum Theorem, <note>angle BAC + angle ABC + angle ACB = 180°</note>.

Definition of Alternate Interior Angles

<note>In the figure of this problem, two parallel lines AD and BC are intersected by a line AC</note>, <note>where angle 1 and angle ACB are located between the two parallel lines and on opposite sides of the intersecting line AC</note>, thus angle 1 and angle ACB are alternate interior angles. <note>Alternate interior angles are equal, that is, angle 1 is equal to angle ACB</note>.


Question: As shown in the figure, the diameter AB of ⊙O is perpendicular to the chord CD, and the foot of the perpendicular is point E. Given ∠CAO = 22.5°, OC = 6, find the length of CD.

Answer: 6√{2}

1. Given AB is the diameter of ⊙O and perpendicular to chord CD, intersecting at point E, according to the perpendicular bisector theorem, CE = DE.

2. ∠CAO = 22.5°, according to the inscribed angle theorem, ∠COB = 2∠CAO = 2×22.5° = 45°.

3. In △OCE, according to the triangle angle sum theorem, ∠OCE = 180° - 90° - 45° = 45°.

4. Since △OCE is an isosceles right triangle, according to the Pythagorean theorem, 2CE^2 = OC^2, CE = 3√2.

5. Since CE = DE, CD = CE + DE = 2×CE = 6√2.

6. Through the above reasoning, the final answer is 6√2.

圆; 垂线; 弦; 直角三角形; 圆周角

Definition of Inscribed Angle

<note>In the figure of this problem, in circle O, the vertex A of angle CAO is on the circumference</note>, <note>the two sides of angle CAO intersect circle O at points C and O respectively</note>. Therefore, <note>angle CAO is an inscribed angle</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>AB is the diameter, connecting the center O and points A and B on the circumference</note>, with a length equal to 2 times the radius, i.e., <note>AB = 2r</note>.

Perpendicular Diameter Theorem

If a diameter is perpendicular to a chord, then that diameter bisects the chord and also bisects the two arcs subtended by that chord.

In the figure of this problem, in circle O, <note>diameter AB is perpendicular to chord CD</note>, then according to the Perpendicular Diameter Theorem, <note>diameter AB bisects chord CD</note>, that is <note>CE=DE</note>, and <note>diameter AB bisects the two arcs subtended by chord CD</note>, that is <note>arc AC = arc AD</note>.

Inscribed Angle Theorem

In the figure of this problem, in circle O, points A, C, and B are on the circle, <note>the central angle corresponding to arc CB and arc CA is ∠COB</note>, <note>the inscribed angle is ∠CAO</note>. According to the Inscribed Angle Theorem, <note>∠CAO is equal to half of the central angle ∠COB corresponding to arc CB</note>, that is, <note>∠CAO = 1/2 ∠COB</note>.

Definition of Isosceles Right Triangle

An isosceles right triangle is a triangle with two sides of equal length and one angle measuring 90 degrees.

In the figure of this problem, triangle OCE is an isosceles right triangle, where <note>angle OEC is a right angle (90 degrees)</note>, <note>sides OE and CE are equal right-angle sides</note>.

Definition of Central Angle

In the figure of this problem, in the circle <note>O</note>, <note>points B and C are two points on the circle</note>, and the center of the circle is <note>point O</note>. The angle ∠BOC formed by the lines <note>OB and OC is called the central angle</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 <note>right triangle OCE</note>, <note>angle OEC is a right angle (90 degrees)</note>, the sides <note>OE and CE are the legs</note>, and the side <note>OC is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>OC² = CE² + OE²</note>.


Question: As shown in the figure, sunlight enters the room through the classroom window. The length of the shadow of the window frame AB on the ground is DE=1.8m, the distance from the bottom of the window to the ground is BC=1m, and EC=1.2m. Then the height of the window AB is ()

Answer: 1.5m

1.## Because BE is parallel to AD, ∠EBC and ∠DAC are corresponding angles, ∠BEC and ∠ADC are corresponding angles, according to the parallel axiom 2 of parallel lines, we can conclude ∠EBC＝∠DAC, ∠BEC＝∠ADC, and according to the similarity theorem (AA), triangle BCE is similar to triangle ACD##.

2. ##From triangle BCE being similar to triangle ACD, according to the definition of similar triangles,## we get CB/AC = CE/CD.

3. According to the problem statement, BC = 1, EC = 1.2, DE = 1.8, substituting these values into the previous proportion formula: BC/(AB + BC) = EC/(DE + EC), that is 1/(AB + 1) = 1.2/(1.8 + 1.2).

4. Calculating the right part of the proportion: 1.2/3 = 0.4.

5. Thus we get the proportion: 1/(AB + 1) = 0.4.

6. Cross multiplying we get: 1 = 0.4 * (AB + 1).

7. Simplifying the equation we get: 1 = 0.4AB + 0.4.

8. Rearranging terms we get: 0.6 = 0.4AB.

10. Therefore, the final height of the window AB is 1.5 meters.

直角三角形; 线段; 平行线; 垂线; 仰角

Definition of Similar Triangles

<note>△BCE is similar to △ACD</note>, according to the definition of similar triangles: <note>∠BCE = ∠ACD</note>, <note>∠CBE = ∠CAD</note>, <note>∠BEC = ∠ADC</note>; <note>BC/AC = CE/CD = BE/AD</note>. Through this proportional relationship, we can relate the sought <note>AB</note> to the other sides of the triangle and determine the value of <note>AB</note> based on other known side lengths.

Similarity Theorem for Triangles (AA)

Two triangles are similar if two of their corresponding angles are congruent.

<note>Angle EBC is equal to angle DAC</note>, and <note>angle BEC is equal to angle ADC</note>, so <note>triangle BCE is similar to triangle ACD</note>.

Parallel Postulate 2 of Parallel Lines

<note>In the diagram of this problem, two parallel lines AD and BE are intersected by a third line AC, forming the following geometric relationship: corresponding angles: angle DAC and angle EBC are equal</note>. These relationships illustrate that when two parallel lines are intersected by a third line, the corresponding angles are equal. <note>Two parallel lines AD and BE are intersected by a third line DC, forming the following geometric relationship: corresponding angles: angle BEC and angle ADC are equal</note>. These relationships illustrate that when two parallel lines are intersected by a third line, the corresponding angles are equal.

Definition of Corresponding Angles

Two angles are referred to as corresponding angles if they are formed when two lines, a and b, are intersected by a third line, c, and both angles lie on the same side of the intersecting line c and on the same side of the lines a and b.

<note>In the diagram of this problem, two parallel lines AD and BE are intersected by a line AC</note>, where <note>angle EBC and angle DAC are on the same side of the intersecting line AC, on the same side of the two intersected lines AD and BE</note>, therefore <note>angle EBC and angle DAC are corresponding angles</note>. Corresponding angles are equal, <note>that is, angle EBC is equal to angle DAC</note>. <note>Two parallel lines AD and BE are intersected by a line DC</note>, where <note>angle BEC and angle ADC are on the same side of the intersecting line DC, on the same side of the two intersected lines AD and BE</note>, therefore <note>angle BEC and angle ADC are corresponding angles</note>. Corresponding angles are equal, <note>that is, angle BEC is equal to angle ADC</note>.


Question: As shown in the figure, a sector-shaped iron sheet OAB is given, with OA=30cm and ∠AOB=120°. The worker folds OA and OB together to form a conical chimney cap (ignoring the seam). Find the radius of the base circle of the chimney cap.

Answer: 10cm

1. Given OA=30 cm, ∠AOB=120°.

2. According to the conversion formula between degrees and radians, convert the central angle to radians to get θ = 120° × (π/180°) = (2π/3).

3. According to the formula for the arc length of a sector L = θr (where L is the arc length, θ is the central angle in radians, and r is the radius), the arc length L = (2π/3) × 30 = 20π cm.

4. Let the radius of the base circle of the chimney cap be r, then according to the formula for the circumference of a circle, the circumference of the base circle is 2πr. Since the worker combined OA and OB to form a conical chimney cap, according to the development diagram of the cone, the arc length of the sector is equal to the circumference of the base circle of the conical cap, so 2πr=20π.

5. Divide both sides by 2π, solving for r=10.

6. Through the above reasoning, the final answer is 10 cm.

Definition of Sector

A sector is a figure formed by two radii of a circle and the arc enclosed between them.

In the diagram of this problem, <note>sector OAB</note>, <note>radius OA</note> and <note>radius OB</note> are two radii of the circle, and <note>arc AB</note> is the arc enclosed by these two radii. Therefore, according to the definition of a sector, the figure composed of these two radii and the enclosed <note>arc AB</note> is a sector. It is known that <note>OA=30 cm</note>, <note>OB=30 cm</note>, and <note>∠AOB=120°</note>.

Circumference Formula of Circle

The circumference of a circle is the length around the circular boundary. It can be calculated using the formula \( C = 2\pi r \), where \( C \) represents the circumference and \( r \) represents the radius.

<note>Let the radius of the circular base of the chimney cap be r</note>, then <note>the circumference of the circular base is 2πr</note>. According to the problem conditions, <note>the arc length of sector OAB is equal to the circumference of the circular base of the conical cap</note>, that is, <note>2πr = 20π</note>.

Formula for the Length of an Arc of a Sector

The length \( L \) of the arc of a sector is equal to the central angle \( \theta \) (measured in radians) multiplied by the radius \( r \): \( L = \theta r \).

In <note>sector OAB</note>, <note>the central angle ∠AOB is 120° (expressed in radians as 2π/3)</note>, <note>and the radius OA is 30 cm</note>. According to the formula for the length of an arc of a sector, the arc length L is equal to the central angle θ multiplied by the radius r, i.e., <note>L = θ * r</note>. The calculated arc length is <note>(2π/3) × 30</note> = 20π cm.

Definition of Radius

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

The length of the line segment from <note>the center of the sector</note> to <note>any point on the circumference</note> is <note>6 centimeters</note>, therefore this <note>line segment is the radius of the sector</note>.

Radians = Degrees × (π/180), Degrees = Radians × (180/π)

<note>In the figure OAB, the angle AOB is 120°</note>, according to the <note>Formula for Conversion between Degrees and Radians</note>, converted to <note>radians = 120° * (π/180)</note>.

Definition of Central Angle

An angle formed by two radii connecting two points on the circumference of a circle to the center of the circle is called a central angle.

<note>In the diagram of this problem, in circle O, points A and B are two points on the circle, and the center of the circle is point O</note>. <note>The angle ∠AOB formed by the lines OA and OB is called the central angle, and the central angle measures 120°</note>.

Cone

A cone is a geometric figure with a circular base and a single vertex. Its surface consists of a curved lateral surface extending from the base to the vertex.

<note>The base of the cone is a circle</note>. <note>The vertex of the cone is point O</note>, <note>the distance between the vertex O and the center of the circle is the height of the cone</note>. <note>The lateral surface of the cone is a curved surface</note>, <note>the distance from vertex O to any point on the circumference is 6 cm</note>.

Development of a Cone

The development (or net) of a cone is a sector of a circle, where the radius of the sector is the slant height of the cone, and the arc length of the sector is equal to the circumference of the cone's base.

<note>The development of a cone is a sector</note>. <note>The radius of the sector is the slant height OA of the cone</note>, <note>and the arc length of the sector is the circumference of the base circle of the cone</note>.


Question: (3 points) As shown in the figure, EF∥BC, AC bisects ∠BAF, ∠B=50°, then the degree of ∠C is ()

Answer: 65°

1. Given EF∥BC, according to the definition of the same-side interior angles, ∠BAF and ∠B are same-side interior angles.

2. According to the parallel axiom 2 of parallel lines, it follows that ∠BAF + ∠B = 180°, i.e., ∠BAF = 180° - ∠B = 180° - 50° = 130°.

3. Given AC bisects ∠BAF, then ∠CAF = ∠BAC, and since ∠CAF + ∠BAC = ∠BAF, it follows that ∠CAF = ∠BAC = ∠BAF / 2 = 130° / 2 = 65°.

4. Since ∠CAF and ∠C are alternate interior angles, therefore ∠C = ∠CAF = 65°.

5. Through the above reasoning, the final answer is 65°.

平行线; 内错角; 同位角; 普通三角形; 三角形的外角

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 angle BAF is point A</note>, from <note>point A, a line AC is drawn</note>, this line <note>divides angle BAF into two equal angles</note>, namely <note>angle BAC and angle CAF are equal</note>. Therefore, <note>line AC is the angle bisector of angle BAF</note>.

Parallel Postulate 2 of Parallel Lines

If two parallel lines are cut by a transversal, then the corresponding angles are equal, the alternate interior angles are equal, and the consecutive interior angles on the same side of the transversal are supplementary.

In the figure of this problem, EF ∥ BC, intersected by lines AB and AC, forming the following geometric relationships: ##1. Corresponding angles: None. 2. Alternate interior angles: <note>∠CAF and ∠C are equal</note>. 3. Consecutive interior angles: <note>∠BAF and ∠B are supplementary, i.e., ∠BAF + ∠B = 180 degrees</note>. These relationships illustrate that when two parallel lines <note>EF and BC are intersected by lines AB and AC</note>, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.

Definition of Consecutive Interior Angles

When two lines are intersected by a third line, the angles that lie on the same side of the transversal and inside the two intersected lines are called consecutive interior angles. If the two lines are parallel, then the consecutive interior angles are supplementary. Conversely, if the consecutive interior angles are supplementary, then the two lines are parallel.

<note>Two lines EF and BC are intersected by a third line AB</note>, <note>angles ∠B and ∠BAF are on the same side of the intersecting line AB and inside the intersected lines EF and BC</note>, so <note>∠B and ∠BAF are consecutive interior angles</note>. Consecutive interior angles <note>∠B and ∠BAF are supplementary</note>, that is, <note>∠BAF + ∠B = 180 degrees</note>.

Definition of Alternate Interior Angles

When a straight line (referred to as a transversal) intersects two parallel lines, the angles that are located between the two parallel lines but on opposite sides of the transversal are called alternate interior angles.

<note>Two parallel lines EF and BC</note> are intersected by <note>a line AC</note>, where <note>angle FAC and angle C are between the two parallel lines</note> and on <note>opposite sides of the intersecting line AC</note>. Therefore, <note>angle FAC and angle C are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>angle FAC is equal to angle C</note>.


Question: As shown in the figure, CD is the diameter of ⊙O, and the chord AB intersects CD at point M. M is the midpoint of AB. Point P is on AD, and PC intersects AB at point N. ∠PNA = 60°, then ∠PDC equals ()

Answer: 60°

1. Given that CD is the diameter of ⊙O, according to the theorem that the angle subtended by a diameter is a right angle, we get ∠CPD = 90°.

2. Since M is the midpoint of AB and CD is the diameter of ⊙O, CM is perpendicular to AB. According to the perpendicular bisector theorem of the circle, CM⊥AB, thus ∠CMA = 90°.

3. Given ∠PNA = 60°, according to the definition of vertical angles, we get ∠CNM = ∠PNA = 60°.

4. Since ∠CMA = 90°, based on the reasoning in step 2, combined with the given condition in step 3 and the angle addition relationship, we get ∠C = 90° - ∠CNM = 90° - 60° = 30°.

5. Adding the auxiliary line PC, thus in △PCD, since ∠CPD = 90°, using the triangle angle sum theorem for right triangles, ∠PDC = 90° - ∠PCD.

6. Therefore, ∠PDC = 90° - ∠PCD = 90° - 30° = 60°.

n. Through the above reasoning, the final answer is ∠PDC = 60°.

圆; 圆周角; 弦; 中点; 直角三角形

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.

In the figure of this problem, <note>CD is the diameter</note>, connecting <note>the center O</note> and <note>points C and D on the circumference</note>, with a length of <note>twice the radius</note>, that is, <note>CD = 2r</note>.

Definition of Inscribed Angle

An inscribed angle is an angle whose vertex lies on the circumference of a circle and whose sides intersect the circle at two points.

In the figure of this problem, <note>in circle O</note>, <note>the vertex P of angle CPD is on the circumference</note>, <note>the two sides of angle CPD intersect circle O at points C and D respectively</note>. Therefore, <note>angle CPD is an inscribed angle</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, in triangle <note>PCD</note>, angle <note>CPD</note> is a right angle (90 degrees), so triangle <note>PCD</note> is a right triangle. Side <note>PC</note> and side <note>CD</note> are the legs, and side <note>PD</note> is the hypotenuse. In triangle <note>CMN</note>, angle <note>CMN</note> is a right angle (90 degrees), so triangle <note>CMN</note> is a right triangle. Side <note>CM</note> and side <note>MN</note> are the legs, and side <note>CN</note> is the hypotenuse.

Midpoint of a Line Segment

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

In the figure of this problem, <note>the midpoint of line segment AB is point M</note>. According to the definition of the midpoint of a line segment, <note>point M divides line segment AB into two equal parts</note>, that is, <note>the lengths of line segments AM and MB are equal</note>. That is, <note>AM = MB</note>.

Corollary to the Inscribed Angle Theorem 2: The Angle Subtended by the Diameter

The angle subtended by the diameter of a semicircle is a right angle; conversely, if an inscribed angle is 90°, then the chord subtending that angle is a diameter.

In the diagram of this problem, in circle O, the <note>diameter CD</note> subtends a <note>right angle ∠CPD (90 degrees)</note>.

Perpendicular Diameter Theorem

If a diameter is perpendicular to a chord, then that diameter bisects the chord and also bisects the two arcs subtended by that chord.

<note>In circle O, diameter CD is perpendicular to chord AB</note>, then according to the Perpendicular Diameter Theorem, <note>diameter CD bisects chord AB, that is, AM=MB</note>, and <note>diameter CD bisects the arcs corresponding to chord AB, that is, arc AC=arc CB</note>.

Definition of Vertical Angles

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

<note>Two intersecting lines AB and CP intersect at point N</note>, forming four angles: <note>angle ANC, angle CNM, angle PNM, and angle ANP</note>. According to the definition of vertical angles, <note>angle ANC and angle PNM are vertical angles</note>, <note>angle CNM and angle ANP are vertical angles</note>. Since vertical angles are equal, <note>angle ANC = angle PNM, angle CNM = angle ANP</note>.

Complementary Acute Angles in a Right Triangle

In a right triangle, the sum of the two non-right angles is 90°.

In the diagram of this problem, in the right triangle CPD, <note>angle CPD is a right angle (90 degrees)</note>, angle C and angle D are the two acute angles other than the right angle. According to the complementary acute angles property of a right triangle, <note>the sum of angle C and angle D is 90 degrees, i.e., angle C + angle D = 90°</note>. In the right triangle CMN, <note>angle CMN is a right angle (90 degrees)</note>, angle C and angle CNM are the two acute angles other than the right angle. According to the complementary acute angles property of a right triangle, <note>the sum of angle C and angle CNM is 90 degrees, i.e., angle C + angle CNM = 90°</note>.


Question:
What is m $\angle $ TSU?

m $\angle $ TSU= $\Box$ °

Answer: m \$\angle \$ TSU=57°

1. In the circle, angle V is the central angle, the central angle is ∠TVU, from the figure we know ∠TVU = 114°.

2. Angle TSU is an inscribed angle, an inscribed angle is an angle whose vertex is on the circle and whose sides intersect the circle.

3. According to the definition of inscribed angle: In a circle, an inscribed angle is equal to half of the central angle that subtends the same arc. Specifically, the inscribed angle ∠TSU intercepts the same arc as the central angle ∠TVU.

4. By the inscribed angle theorem, we have ∠TSU = 1/2 * ∠TVU.

5. Substituting the known ∠TVU = 114°, we get ∠TSU = 1/2 * 114°.

6. Performing the calculation, 1/2 * 114° = 57°.

7. Through the above reasoning, the final answer is 57°.

Definition of Central Angle

<note>Point T and point U are two points on the circle</note>, <note>the center of the circle is point V</note>. The angle ∠TVU formed by the lines <note>VT and VU</note> is called the central angle, and it is known that <note>∠TVU = 114°</note>.

Definition of Inscribed Angle

The vertex S of angle TSU is on the circumference of the circle, and the two sides of angle TSU intersect the circle at points T and U respectively. Therefore, <note>angle TSU is an inscribed angle</note>.

Inscribed Angle Theorem

In the figure of this problem, in circle V, points U, T, and S are on the circle, <note>the central angle corresponding to arc UT is ∠TVU</note>, <note>the inscribed angle is ∠TSU</note>. According to the Inscribed Angle Theorem, <note>∠TSU is equal to half of the central angle ∠TVU corresponding to arc TU</note>, that is, <note>∠TSU = 1/2 ∠TVU</note>.


Question:
The diagram shows a convex polygon.

What is the sum of the exterior angle measures, one at each vertex, of this polygon?
 $\Box$ °

Answer: 360°

1. Observe the given convex polygon and define the exterior angle at each vertex. An exterior angle is the angle formed between the extension of one side of the polygon and the adjacent side.

2. According to the definition of exterior angles, if the polygon has n vertices, there will be n exterior angles.

3. According to the exterior angle sum theorem, the sum of the exterior angles of any convex polygon, regardless of the number of sides, is always equal to 360°, independent of the number of sides of the polygon.

4. This is because at each vertex of the polygon, the interior angle and the exterior angle form an angle of 180°, and the sum of the interior angles of the polygon can be calculated using the interior angle sum theorem, with the formula (n-2) × 180°.

5. Using the exterior angle sum theorem, in this problem, the sum of the exterior angles of the convex polygon is 360°.

6. Therefore, there is no need to use specific shape information of the polygon, directly apply the principle of the exterior angle sum.

7. Through the above reasoning, the final answer is 360°.

Exterior Angle Sum Theorem of Polygon

For any polygon, the sum of its exterior angles is equal to 360°.

Original: <note>The sum of the exterior angles at each vertex of the convex polygon is always 360°</note>. According to the Exterior Angle Sum Theorem of Polygon, <note>the sum of these exterior angles is equal to 360°</note>.

Definition of Exterior Angle of a Polygon

An exterior angle of a polygon is the angle formed between one side of the polygon and the extension of an adjacent side.

<note>In the figure of this problem, an exterior angle of a polygon is the angle formed by one of its interior angles and the extension of its adjacent side</note>. For example, if a vertex of the polygon is A and the adjacent sides are AB and AD, then the exterior angle of the interior angle <note>∠BAD</note> is <note>∠A'</note>, satisfying <note>∠BAD + ∠A' = 180°</note>.


Question:
In the cube shown below, which lines are parallel? Select all that apply.

-

| $\overleftrightarrow{UY}$ | and | $\overleftrightarrow{WX}$ |
-

| $\overleftrightarrow{VY}$ | and | $\overleftrightarrow{WX}$ |
-

| $\overleftrightarrow{XY}$ | and | $\overleftrightarrow{VW}$ |
-

| $\overleftrightarrow{VW}$ | and | $\overleftrightarrow{WX}$ |

Answer:
-

| \$\overleftrightarrow{VY}\$ | and | \$\overleftrightarrow{WX}\$ |
-

| \$\overleftrightarrow{XY}\$ | and | \$\overleftrightarrow{VW}\$ |

1. Given that the vertices of the cube are R, S, T, U, V, W, X, Y.

2. According to the properties of skew lines, segments UY and WX lie in different planes, i.e., UY and WX lie in planes UTX and STWX respectively. Therefore, these two lines are skew lines and not parallel lines.

3. Segments VY and WX both lie in the plane VWYX. According to the condition that two lines in the same plane do not intersect, these two lines are parallel.

4. Segments XY and VW lie in the same plane VWYX, and according to the definition of parallel lines, two lines in the same plane do not intersect. Therefore, XY and VW are parallel.

5. Segments VW and WX lie in the same plane VWYX and intersect at point W. Therefore, VW and WX are intersecting lines and not parallel lines.

6. Based on the above reasoning, the pairs of parallel lines in the problem are ⟶VY and ⟶WX, as well as ⟶XY and ⟶VW.

Definition of Parallel Lines

Two lines in the same plane that do not intersect are called parallel lines.

The segments <note>VY and WX</note>, as well as <note>XY and VW</note>, are parallel lines because they lie within the same <note>plane VWX</note> and do not intersect. According to the definition of parallel lines, these segments meet the conditions for parallel lines.

Properties of Skew Lines

Two lines are called skew lines if and only if they are not parallel and do not lie in the same plane. Skew lines do not intersect.

In the figure of this problem, <note>line UY and WX are not parallel and not in the same plane, they are skew lines</note>, so <note>UY and WX do not intersect</note>.


Question:
What is m $\angle $ HGI?

m $\angle $ HGI= $\Box$ °

Answer: m \$\angle \$ HGI=37°

1. In the given geometric figure, ∠HGI is an inscribed angle, while ∠J is a central angle, and these two angles subtend the same arc, namely arc HI.

2. According to the inscribed angle theorem, any inscribed angle is equal to half of the central angle that subtends the same arc. Mathematically, it can be described as: if an inscribed angle ∠ABC corresponds to the same arc as the central angle ∠AOC, then m∠ABC = 1/2·m∠AOC.

3. Specifically applied to the problem, we know ∠HGI = 1/2·∠J.

4. Given ∠J = 74°, substituting into ∠HGI, we get: m∠HGI = 1/2 × 74°.

5. Performing the calculation, 1/2 × 74° = 37°.

6. Through the above reasoning, the final answer is ∠HGI = 37°.

Definition of Inscribed Angle

In the problem diagram, in circle O, the vertex G of angle HGI is on the circumference, and the two sides of angle HGI intersect circle O at points H and I respectively. Therefore, angle HGI is an inscribed angle.

Definition of Central Angle

An angle formed by two radii connecting two points on the circumference of a circle to the center of the circle is called a central angle.

<note>Point H and point I are two points on the circle</note>, <note>the center of the circle is point J</note>. The angle ∠HJI formed by the lines <note>JH and JI is called the central angle</note>.

Inscribed Angle Theorem

<note>In circle GHI, points G, H, and I are on the circle</note>, <note>the central angle corresponding to arc HI is ∠HJI</note>, <note>the inscribed angle is ∠HGI</note>. According to the Inscribed Angle Theorem, <note>∠HGI is equal to half of the central angle ∠HJI corresponding to arc HI</note>, that is, <note>∠HGI = 1/2 ∠HJI</note>.


Question:
Find the area of △VWX.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
 $\Box$  mm ^ 2

Answer: 192.0 mm ^ 2

1. Given the side lengths of triangle △VWX are VW=39 mm, WX=17 mm, VX=27 mm.

2. To find the area of the triangle using Heron's formula, we first need to calculate the semi-perimeter s of △VWX. The formula for calculating the semi-perimeter is: s = (VW + WX + VX) / 2.

3. Substitute the given side lengths into the semi-perimeter formula: s = (39 + 17 + 27) / 2 = 83 / 2 = 41.5 mm.

4. Next, calculate the area using Heron's formula. Heron's formula is: Area = √[s(s-VW)(s-WX)(s-VX)].

5. Substitute the semi-perimeter s obtained in step 3 and the given side lengths into Heron's formula: Area = √[41.5 * (41.5-39) * (41.5-17) * (41.5-27)].

6. Perform step-by-step calculations: 41.5-39=2.5, 41.5-17=24.5, 41.5-27=14.5.

7. Continue with the product calculation: 41.5 * 2.5 * 24.5 * 14.5 = 36,857.1875.

8. Finally, calculate the area of the triangle: Area = √36,857.1875 ≈ 191.9822… mm².

9. According to the problem requirements, the final answer needs to be rounded to the nearest tenth, thus rounding to 192.0 mm².

10. Based on the above reasoning, the final answer is 192.0 mm².

Definition of Triangle

A geometric figure is called a triangle if it is formed by three non-collinear points and the line segments connecting them.

<note>Triangle △VWX</note> is a geometric figure composed of <note>three non-collinear points V, W, X and their connecting line segments VW, WX, VX</note>. <note>Points V, W, X are the three vertices of the triangle</note>, and <note>line segments VW, WX, VX are the three sides of the triangle</note>.

Heron's Formula

Heron's formula is used to calculate the area of any triangle. The formula is given by: \( A = \sqrt{s(s - a)(s - b)(s - c)} \), where \( s \) is the semi-perimeter, and \( a, b, \) and \( c \) are the lengths of the sides of the triangle.

In the figure of this problem, use Heron's formula to calculate the area of triangle <note>△VWX</note>. The specific steps are as follows: the known side lengths are <note>a=39 mm (VW)</note>, <note>b=17 mm (WX)</note>, <note>c=27 mm (VX)</note>. First, calculate the semi-perimeter <note>s = 41.5 mm</note>, then substitute into the formula <note>Area = √[41.5 * (41.5-39) * (41.5-17) * (41.5-27)]</note>. Step-by-step calculations yield: 41.5-39=2.5, 41.5-17=24.5, 41.5-27=14.5, performing the product calculation results in <note>√36857.1875 ≈ 191.9822 mm²</note>, rounded to the nearest tenth, therefore the answer is <note>192.0 mm²</note>.


Question:
In the cube shown below, which lines are parallel?

-

| $\overleftrightarrow{RU}$ | and | $\overleftrightarrow{UY}$ |
-

| $\overleftrightarrow{VW}$ | and | $\overleftrightarrow{UY}$ |
-

| $\overleftrightarrow{TX}$ | and | $\overleftrightarrow{WX}$ |
-

| $\overleftrightarrow{TX}$ | and | $\overleftrightarrow{UY}$ |

Answer:
-

| \$\overleftrightarrow{TX}\$ | and | \$\overleftrightarrow{UY}\$ |

Starting from the geometric structure of the cube, first check the lines \\overleftrightarrow{RU} and \\overleftrightarrow{UY}. From the figure, it can be seen that the lines \\overleftrightarrow{RU} and \\overleftrightarrow{UY} intersect at point U, so they are intersecting lines, not parallel lines. According to the definition of parallel lines: two lines are parallel if they lie in the same plane and do not intersect. Therefore, \\overleftrightarrow{RU} and \\overleftrightarrow{UY} are not parallel.

Then check the lines \\overleftrightarrow{VW} and \\overleftrightarrow{UY}. \\overleftrightarrow{VW} is located on \\overline{VWR}, while \\overleftrightarrow{UY} is located on \\overline{UYX}. Since these two lines are not parallel and not in the same plane, they are skew lines. According to the definition of parallel lines, parallel lines must be in the same plane, so \\overleftrightarrow{VW} and \\overleftrightarrow{UY} are not parallel.

Next, check the lines \\overleftrightarrow{TX} and \\overleftrightarrow{WX}. From the structure of the cube, it can be seen that these two lines intersect at the same point X on the same face \\overline{TWX} of the cube. Therefore, \\overleftrightarrow{TX} and \\overleftrightarrow{WX} are intersecting lines, not parallel lines.

Finally, check the lines \\overleftrightarrow{TX} and \\overleftrightarrow{UY}. Observing the structure, it can be found that the lines \\overleftrightarrow{TX} and \\overleftrightarrow{UY} lie in the same plane \\overline{TUX} and do not intersect. According to the definition of parallel lines, they are parallel lines.

After the above reasoning, the final answer is that \\overleftrightarrow{TX} and \\overleftrightarrow{UY} are parallel.

Definition of Parallel Lines

<note>\overleftrightarrow{TX}</note> and <note>\overleftrightarrow{UY}</note> lie in the same plane <note>\overline{TUX}</note> and do not intersect, so according to the definition of parallel lines, <note>\overleftrightarrow{TX}</note> and <note>\overleftrightarrow{UY}</note> are parallel lines. <note>\overleftrightarrow{RU}</note> and <note>\overleftrightarrow{UY}</note> lie in the same plane but intersect at point <note>U</note>, so they are not parallel.

Definition of Skew Lines

Two lines are called skew lines if and only if they are not parallel and do not lie in the same plane. Skew lines are also known as non-coplanar lines.

<note>Line VW and line UY are not parallel and do not lie in the same plane</note>, they are skew lines, also known as non-coplanar lines.


Question:
In the rectangular prism shown below, which lines are parallel? Select all that apply.

-

| $\overleftrightarrow{QU}$ | and | $\overleftrightarrow{ST}$ |
-

| $\overleftrightarrow{UX}$ | and | $\overleftrightarrow{VW}$ |
-

| $\overleftrightarrow{SW}$ | and | $\overleftrightarrow{UV}$ |
-

| $\overleftrightarrow{UV}$ | and | $\overleftrightarrow{WX}$ |

Answer:
-

| \$\overleftrightarrow{UX}\$ | and | \$\overleftrightarrow{VW}\$ |
-

| \$\overleftrightarrow{UV}\$ | and | \$\overleftrightarrow{WX}\$ |

1. Given the figure is a rectangular prism, we need to identify which line segments are parallel. According to the definition of parallel lines, two lines are parallel if they lie in the same plane and never intersect.

2. First, consider the lines \\(ackslashackslashoverleftrightarrow{QU}\\) and \\(ackslashackslashoverleftrightarrow{ST}\\). They belong to two different planes \\(ackslashackslashoverleftrightarrow{QUXT}\\) and \\(ackslashackslashoverleftrightarrow{QRST}\\), so it is directly known that these two lines are not in the same plane and are skew lines. Skew lines cannot be parallel, so \\(ackslashackslashoverleftrightarrow{QU}\\) and \\(ackslashackslashoverleftrightarrow{ST}\\) are not parallel lines.

3. Next, consider the lines \\(ackslashackslashoverleftrightarrow{UX}\\) and \\(ackslashackslashoverleftrightarrow{VW}\\). They are both in the same plane \\(ackslashackslashoverleftrightarrow{UVWX}\\) and are adjacent to other lines in that plane without sharing points, i.e., they are parallel lines.

4. Then, consider the lines \\(ackslashackslashoverleftrightarrow{SW}\\) and \\(ackslashackslashoverleftrightarrow{UV}\\). Among them, \\(ackslashackslashoverleftrightarrow{SW}\\) is in the plane \\(ackslashackslashoverleftrightarrow{SWVR}\\), while \\(ackslashackslashoverleftrightarrow{UV}\\) is in the plane \\(ackslashackslashoverleftrightarrow{UVWX}\\), not in the same plane, and are skew lines, so they are not parallel lines.

5. Finally, consider the lines \\(ackslashackslashoverleftrightarrow{UV}\\) and \\(ackslashackslashoverleftrightarrow{WX}\\). These two lines are in the same plane \\(ackslashackslashoverleftrightarrow{UVWX}\\) and do not intersect, so they are parallel lines.

6. Through the above analysis, we determine that there are two sets of parallel lines: \\(ackslashackslashoverleftrightarrow{UX}\\) and \\(ackslashackslashoverleftrightarrow{VW}\\), and \\(ackslashackslashoverleftrightarrow{UV}\\) and \\(ackslashackslashoverleftrightarrow{WX}\\).

Definition of Parallel Lines

In the diagram for this problem, <note>line segments \(\overleftrightarrow{UX}\) and \(\overleftrightarrow{VW}\) are parallel lines</note> because they lie on the plane \(UVWX\) and do not intersect. Similarly, <note>line segments \(\overleftrightarrow{UV}\) and \(\overleftrightarrow{WX}\) are also parallel lines</note>, lying on the plane \(UVWX\) and do not intersect.

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 <note>combination relationship of faces on a rectangular prism</note> was analyzed in detail, determining that <note>plane \(UVWX\) is parallel to plane \(QRST\)</note>, and that <note>plane \(QUXT\) is parallel to plane \(SWVR\)</note>. This allowed the use of the <note>properties of parallel planes in a rectangular prism</note> when determining the relationships between line segments.


Question:
The diagram shows a convex polygon.

What is the sum of the exterior angle measures, one at each vertex, of this polygon?
 $\Box$ °

Answer: 360°

1. Given that this is a convex polygon, the exterior angle of a polygon is defined as: the angle formed by one interior angle of the polygon and the extension of the adjacent side is called the exterior angle of this interior angle.

2. According to the exterior angle sum theorem of polygons, this theorem states: the sum of the exterior angles of any polygon is equal to 360°.

3. For any shape of convex polygon, this conclusion still holds, regardless of how many sides the polygon has.

4. Observe the given polygon in the figure, confirm that it is a convex polygon, thereby verifying that the exterior angle sum theorem of polygons can be applied.

5. Hence, the sum of the exterior angles at each vertex in the figure is equal to 360°.

6. Through the above reasoning, the final answer is 360°.

In the diagram of this problem, <note>an interior angle of the polygon</note>, <note>the angle formed by extending the adjacent side of this interior angle</note> is called the <note>exterior angle</note> of this interior angle.

Exterior Angle Sum Theorem of Polygon

Original: <note>In the convex polygon in the figure of this problem</note>, the sum of all exterior angles is 360° according to the <note>Exterior Angle Sum Theorem of Polygon</note>. According to the Exterior Angle Sum Theorem of Polygon, the <note>sum of these exterior angles is equal to 360°</note>.


Question:
What is m $\angle $ EDF?

m $\angle $ EDF= $\Box$ °

Answer: m \$\angle \$ EDF=31°

1. Given that angle G is a central angle, and ∠EDF is an inscribed angle, and the arc corresponding to ∠EDF is the same as the arc corresponding to the central angle ∠EGF.

2. According to the inscribed angle theorem, which states: an inscribed angle in a circle is equal to half of the central angle that subtends the same arc. Therefore, ∠EDF = 1/2 × ∠EGF.

3. Given in the problem: ∠EGF = 62°, substituting into the inscribed angle theorem gives: ∠EDF = 1/2 × 62°.

4. Calculating ∠EDF = 31°.

5. Through the above reasoning, the final answer is 31°.

Definition of Central Angle

In the figure of this problem, in the circle, <note>point E and point F are two points on the circle</note>, <note>the center of the circle is point G</note>. <note>The angle ∠EGF formed by the lines GE and GF is called the central angle</note>.

Definition of Inscribed Angle

An inscribed angle is an angle whose vertex lies on the circumference of a circle and whose sides intersect the circle at two points.

In the <note>circle</note>, the vertex D of angle EDF is <note>on the circumference</note>, and the two sides of angle EDF <note>intersect the circle at points E and F</note> respectively. Therefore, angle EDF is an <note>inscribed angle</note>.

Inscribed Angle Theorem

<note>In circle G, points E, F, D are on the circle</note>, the central angle corresponding to arc EF is <note>∠EGF</note>, and the inscribed angle is <note>∠EDF</note>. According to the Inscribed Angle Theorem, <note>∠EDF is equal to half of the central angle ∠EGF corresponding to arc EF</note>, i.e., <note>∠EDF = 1/2 ∠EGF</note>.


Question:
What is m $\angle $ H?

m $\angle $ H= $\Box$ °

Answer: m \$\angle \$ H=146°

1. Observing the figure, ∠JIK is an inscribed angle, and the arc JK it intercepts is the same as the central angle ∠KHJ.

2. According to the Inscribed Angle Theorem, an inscribed angle is equal to half of its intercepted central angle. In this problem, ∠JIK is the inscribed angle, and ∠KHJ is the central angle.

3. The Inscribed Angle Theorem states that an inscribed angle is equal to half of its corresponding central angle. Therefore, ∠KHJ = 2 × ∠JIK.

4. Given that ∠JIK = 73°, substitute into the formula to get ∠KHJ = 2 × 73°.

5. Calculating, we get ∠KHJ = 146°.

6. Based on the above reasoning, the final answer is 146°.

Definition of Inscribed Angle

<note>In the diagram of this problem, in circle H</note>, the vertex of angle JIK <note>I is on the circumference</note>, and the two sides of angle JIK intersect <note>circle H at points J and K</note>. Therefore, <note>angle JIK is an inscribed angle</note>.

Definition of Central Angle

In the diagram of this problem, in the circle, point <note>K</note> and point <note>J</note> are two points on the circle, <note>the center of the circle is point H</note>. The angle ∠<note>KHJ</note> formed by the lines <note>HK</note> and <note>HJ</note> is called the central angle.

Inscribed Angle Theorem

<note>Points J, I, K are on the circle</note>, <note>the central angle corresponding to arc JK is ∠KHJ</note>, <note>the inscribed angle is ∠JIK</note>. According to the Inscribed Angle Theorem, <note>∠JIK is equal to half of the central angle ∠KHJ corresponding to arc JK</note>, that is, <note>∠JIK = 1/2 ∠KHJ</note>.


Question:
What is m $\angle $ EGF?

m $\angle $ EGF= $\Box$ °

Answer: m \$\angle \$ EGF=33°

1. Given a circle, ∠EGF is an inscribed angle, and it intercepts the same arc EF as the central angle ∠EHF.

2. According to the Inscribed Angle Theorem, the inscribed angle ∠EGF is equal to half of the central angle ∠EHF that intercepts the same arc.

3. Given that the measure of the central angle ∠EHF is 66°, according to the theorem we have: ∠EGF = 1/2 × m ∠EHF.

4. Substituting ∠EHF = 66°, we get: ∠EGF = 1/2 × 66° = 33°.

5. Through the above reasoning, the final answer is m ∠EGF = 33°.

Definition of Inscribed Angle

<note>In circle O, the vertex G of angle EGF is on the circumference</note>, <note>the two sides of angle EGF intersect circle O at points E and F respectively</note>. Therefore, <note>angle EGF is an inscribed angle</note>.

Definition of Central Angle

In the diagram of this problem, in the circle, <note>point E and point F are two points on the circle</note>, <note>the center of the circle is point H</note>. <note>The angle ∠EHF formed by the lines HE and HF is called the central angle</note>.

Inscribed Angle Theorem

<note>∠EGF is an inscribed angle</note>, corresponding <note>central angle is ∠EHF</note>, according to the Inscribed Angle Theorem, <note>∠EGF = 1/2 × ∠EHF</note>.


Question:
In the triangular prism shown below, which lines are intersecting?

-

| $\overleftrightarrow{ST}$ | and | $\overleftrightarrow{WX}$ |
-

| $\overleftrightarrow{UX}$ | and | $\overleftrightarrow{VW}$ |
-

| $\overleftrightarrow{SU}$ | and | $\overleftrightarrow{TU}$ |
-

| $\overleftrightarrow{TU}$ | and | $\overleftrightarrow{WX}$ |

Answer:
-

| \$\overleftrightarrow{SU}\$ | and | \$\overleftrightarrow{TU}\$ |

1. First, observe the line segments in the three-dimensional geometric figure, which is a triangular prism.

2. Consider the line segments $\overleftrightarrow{ST}\$ and $\\overleftrightarrow{WX}\\$. These two line segments are on different planes, where $\\overleftrightarrow{ST}\\$ is on the top and side planes, while $\\overleftrightarrow{WX}\\$ is on the bottom plane. Since they are on different planes, according to the definition of skew lines, $\\overleftrightarrow{ST}\\$ and $\\overleftrightarrow{WX}\\$ are skew lines. Based on the properties of skew lines, skew lines do not intersect.

3. Consider the line segments $\\overleftrightarrow{UX}\\$ and $\\overleftrightarrow{VW}\\$. These two line segments are also on different planes, where $\\overleftrightarrow{UX}\\$ is the line segment connecting U and X, and $\\overleftrightarrow{VW}\\$ is on the bottom plane. Since they are not on the same plane, according to the properties of skew lines, they do not intersect.

4. Consider the line segments $\\overleftrightarrow{SU}\\$ and $\\overleftrightarrow{TU}\\$. These two line segments share point U and are on the same plane. According to the definition of two lines intersecting on a plane in plane geometry, these two line segments intersect at point U.

5. Consider the line segments $\\overleftrightarrow{TU}\\$ and $\\overleftrightarrow{WX}\\$. The line segment $\\overleftrightarrow{TU}\\$ and $\\overleftrightarrow{WX}\\$ are on the same plane and do not share any common points. According to the definition of parallel lines, the two lines do not intersect.

6. Based on the above analysis, the only intersecting line segments are $\\overleftrightarrow{SU}\\$ and $\\overleftrightarrow{TU}\\$, which intersect at point U.

7. In conclusion, based on the properties and analysis of the geometric figure, we conclude that $\\overleftrightarrow{SU}\\$ and $\\overleftrightarrow{TU}\\$ are the only intersecting line segments in the problem.

Definition of Parallel Lines

In the figure of this problem, <note>\$\overleftrightarrow{TU}\$ and \$\overleftrightarrow{WX}\$</note> lie in the same plane \overline{TUX} and they have no intersection points, so according to the definition of parallel lines, <note>\$\overleftrightarrow{TU}\$ and \$\overleftrightarrow{WX}\$ are parallel lines.</note>

Definition of Skew Lines

<note>\$\overleftrightarrow{ST}\$ and \$\overleftrightarrow{WX}\$ are not parallel and not in the same plane, they are skew lines, also known as non-coplanar lines</note>; <note>\$\overleftrightarrow{UX}\$ and \$\overleftrightarrow{VW}\$ are not parallel and not in the same plane, they are skew lines, also known as non-coplanar lines</note>.

Properties of Skew Lines

In this problem diagram, <note>line segment $\overleftrightarrow{ST}$ and $\overleftrightarrow{WX}$ are not parallel and not in the same plane, they are skew lines</note>, so <note>$\overleftrightarrow{ST}$ and $\overleftrightarrow{WX}$ do not intersect</note>. <note>Line segment $\overleftrightarrow{UX}$ and $\overleftrightarrow{VW}$ are not parallel and not in the same plane, they are skew lines</note>, so <note>$\overleftrightarrow{UX}$ and $\overleftrightarrow{VW}$ do not intersect</note>. <note>Line segment $\overleftrightarrow{TU}$ and $\overleftrightarrow{WX}$ are not parallel and not in the same plane, they are skew lines</note>, so <note>$\overleftrightarrow{TU}$ and $\overleftrightarrow{WX}$ do not intersect</note>.


Question:
Find the lateral area of the prism. The base is an equilateral triangle.

 $\Box$  square centimeters

Answer: 900 square centimeters

1. Given that the base of the prism is an equilateral triangle, each side is 15 cm. According to the definition of an equilateral triangle, its three sides are equal, so the perimeter of the base can be expressed as three times the length of each side.

2. Using the perimeter formula for a triangle, since the sides of an equilateral triangle are equal, the perimeter p = 3s, where s represents the length of one side of the equilateral triangle.

3. Substituting the given side length s = 15, we get the perimeter of the base p = 3 × 15 = 45 cm.

4. The lateral area of the prism is obtained by multiplying the perimeter of the base by the height of the prism. The formula for the lateral area is L.A. = p × h, where h represents the height of the prism.

5. Given the height of the prism h = 20 cm, substituting into the formula gives L.A. = 45 × 20 = 900.

6. Through the above calculations, the lateral area is 900 square centimeters.

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 base of the prism is an equilateral triangle (two parallel and congruent triangular bases)</note>, <note>the lateral faces are rectangles</note>. The height of the prism is 20 centimeters.

Lateral Surface Area Formula of a Prism

The lateral surface area of a prism is equal to the perimeter of the base multiplied by the height.

<note>The perimeter of the base equilateral triangle p = 45 cm</note> and <note>the height of the prism h = 20 cm</note>, <note>the lateral surface area L.A. = 45 × 20 = 900 square cm</note>.

Formula for the Perimeter of a Triangle

The perimeter of a triangle is the sum of the lengths of its three sides. Therefore, if the three sides of the triangle are denoted as \( a \), \( b \), and \( c \), then the perimeter \( P \) is given by \( P = a + b + c \).

In the figure of this problem, in the triangle, <note>the three sides are 15, 15, 15 respectively</note>, according to the formula for the perimeter of a triangle, which is <note>Perimeter L = 15 + 15 + 15</note>.


Question:
Find the lateral area of the triangular prism.

 $\Box$  square centimeters

Answer: 1,536 square centimeters

1. Confirm that the base shape of the triangular prism is a right triangle with side lengths of 12 cm, 16 cm, and 20 cm. According to the converse of the Pythagorean theorem c² = a² + b², verify that a=12 cm, b=16 cm, and c=20 cm satisfy 12² + 16² = 20², thus it is a right triangle.

2. According to the problem statement, the lateral area of the triangular prism is the product of the base perimeter and the prism height. First, calculate the perimeter of the right triangle: 12 + 16 + 20 = 48 cm.

3. The problem states that the height of the triangular prism is 32 cm. Substitute into the formula to calculate the lateral area: 48 (perimeter) multiplied by 32 (height).

4. Perform the multiplication: 48 * 32 = 1536 square cm.

5. Through the above reasoning, the final answer is 1536 square cm.

Definition of Right Triangle

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

<note>The base shape is a right triangle with side lengths of 12 cm, 16 cm, and 20 cm</note>. <note>The sides with lengths of 12 cm and 16 cm are the legs</note>, <note>and the side with a length of 20 cm is the hypotenuse</note>.

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>The base of the triangular prism is a right triangle with side lengths of 12 cm, 16 cm, and 20 cm</note>, and the height of the triangular prism is 32 cm.

Converse of the Pythagorean Theorem

If the sum of the squares of two sides of a triangle is equal to the square of the third side, then the triangle is a right triangle, and the angle opposite to the longest side is a right angle.

<note>In the given problem diagram, it is known that the base of the triangular prism has side lengths of a=12 cm, b=16 cm, and c=20 cm,</note><note>and satisfies c² = a² + b²,</note><note>then according to the converse of the Pythagorean Theorem, the base is a right triangle, and the angle opposite the longest side c is a right angle.</note>

Lateral Surface Area Formula of a Prism

<note>Base perimeter 12 + 16 + 20 = 48 cm</note> multiplied by <note>prism height 32 cm</note> gives the lateral surface area <note>48 * 32 = 1536 square cm</note>.


Question:
What is m $\angle $ EGF?

m $\angle $ EGF= $\Box$ °

Answer: m \$\angle \$ EGF=66°

1. Given four points E, G, F on a circle and the center H, as well as the inscribed angle ∠EGF, and the central angle ∠EHF = 132°.

2. According to the Inscribed Angle Theorem, the inscribed angle is equal to half of the central angle that subtends the same arc. In this problem, ∠EGF is the inscribed angle, and ∠EHF is the corresponding central angle.

3. Applying the Inscribed Angle Theorem: ∠EGF = 1/2 × ∠EHF.

4. Substitute the given ∠EHF = 132° into the Inscribed Angle Theorem formula: ∠EGF = 1/2 × 132°.

6. Through the above reasoning, the final answer is ∠EGF = 66°.

Definition of Inscribed Angle

<note>The vertex G of angle ∠EGF is on the circumference</note>, and the two sides of angle ∠EGF intersect the circle at points E and F respectively. Therefore, <note>angle ∠EGF is an inscribed angle</note>.

Definition of Central Angle

In the figure of this problem, in the circle, <note>point E and point F are two points on the circle</note>, <note>the center of the circle is point H</note>. <note>The angle ∠EHF formed by the lines HE and HF</note> is called the central angle.

Inscribed Angle Theorem

In a circle, an inscribed angle is equal to half of the central angle that subtends the same arc.

In the figure of this problem, there are points <note>E, G, F on the circle</note>, <note>the central angle corresponding to arc EF is ∠EHF</note>, <note>the inscribed angle is ∠EGF</note>. According to the Inscribed Angle Theorem, <note>∠EGF is equal to half of the central angle ∠EHF corresponding to the arc EF</note>, that is, <note>∠EGF = 1/2 ∠EHF</note>.


Question:
In the cube shown below, which lines are skew?

-

| $\overleftrightarrow{GK}$ | and | $\overleftrightarrow{IM}$ |
-

| $\overleftrightarrow{IM}$ | and | $\overleftrightarrow{GH}$ |
-

| $\overleftrightarrow{HI}$ | and | $\overleftrightarrow{GH}$ |
-

| $\overleftrightarrow{FJ}$ | and | $\overleftrightarrow{JM}$ |

Answer:
-

| \$\overleftrightarrow{IM}\$ | and | \$\overleftrightarrow{GH}\$ |

1. Define skew lines in space. According to the definition of skew lines, skew lines are lines that lie in different planes and do not intersect.

2. Consider the lines \( \overleftrightarrow{GK} \) and \( \overleftrightarrow{IM} \). They lie in the same plane \( (GIMK) \), so they are not skew lines.

3. Consider the lines \( \overleftrightarrow{IM} \) and \( \overleftrightarrow{GH} \). The line \( \overleftrightarrow{IM} \) lies in the plane \( (IMLH) \), and the line \( \overleftrightarrow{GH} \) lies in the plane \( (GHLK) \). They do not lie in the same plane and do not intersect, so they are skew lines.

4. Consider the lines \( \overleftrightarrow{HI} \) and \( \overleftrightarrow{GH} \). They intersect at point H, so they are not skew lines.

5. Consider the lines \( \overleftrightarrow{FJ} \) and \( \overleftrightarrow{JM} \). They intersect at point J, so they are not skew lines.

6. Based on the above analysis, the final answer is that \( \overleftrightarrow{IM} \) and \( \overleftrightarrow{GH} \) are skew lines.

Definition of Skew Lines

<note>In the figure of this problem, the lines xx and xx are not parallel and not in the same plane, they are skew lines, also known as oblique lines.</note>


Question:
Find the lateral area of the cylinder.

Round your answer to the nearest tenth.

 $\Box$  square meters

Answer: 804.2 square meters

1. Given the height of the cylinder is 16 meters and the diameter is 16 meters.

2. Based on the diameter, the radius of the cylinder can be obtained as half of the diameter, i.e., 16 ÷ 2 = 8 meters.

3. The formula for calculating the lateral area of the cylinder is: Lateral Area = 2 × 𝜋 × radius × height.

4. Substitute the known radius and height into the formula: Lateral Area = 2 × 𝜋 × 8 × 16.

5. Perform the specific calculation: Lateral Area = 2 × 3.14159265 × 8 × 16.

7. Round to the nearest tenth: Lateral Area ≈ 804.2.

8. Through the above reasoning, the final answer is 804.2 square meters.

Formula for Lateral Area of a Cylinder

The lateral area (L.A.) of a cylinder is calculated using the formula L.A. = 2πrh, where r is the radius of the base, h is the height, and π represents the constant Pi, which is the ratio of the circumference of a circle to its diameter.

Substitute the <note>radius of the cylinder</note> and <note>height</note> into the formula to calculate the lateral area: Side Area = 2 × 𝜋 × <note>8 meters</note> × <note>16 meters</note>. Finally, the calculated approximate value of the lateral area is 804.2 square meters.


Question:
Find the lateral area of the cone.

Round your answer to the nearest tenth.

 $\Box$  square centimeters

Answer: 763.4 square centimeters

1. Given that the slant height (𝓁) of the cone is 27 cm, and the diameter (d) is 18 cm.

2. The radius of the base of the cone (r) is half of the diameter, i.e., r = d/2 = 18/2 = 9 cm.

3. According to the formula for the lateral area of the cone, the lateral area (L.A.) = πr𝓁.

4. Substituting the given base radius and slant height into the formula, we get: L.A. = π * 9 * 27.

5. Calculating π * 9 * 27, we get L.A. = 763.4070… square cm.

6. According to the rounding principle, the result is rounded to the first decimal place, giving approximately 763.4 square cm.

7. Through the above reasoning, the final answer is 763.4 square cm.

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).

The <note>slant height of the cone (𝓁)</note> is 27 cm, the <note>diameter of the base circle (d)</note> is 18 cm, the apex is directly above the center of the base circle of the cone, where the <note>radius of the base circle (r)</note> is half of the diameter, i.e., <note>r = 9 cm</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.

In the figure of this problem, it is known that <note>the radius (r) of the base circle is 9 cm</note>, <note>the slant height (𝓁) is 27 cm</note>, therefore, according to the formula <note>L.A. = π * 9 * 27</note> to calculate the lateral surface area of the cone.


Question:
Find the lateral area of the prism. The base is a square.

 $\Box$  square feet

Answer: 1,344 square feet

1. Given that the base of the prism is a square, according to the definition of a square, all sides of a square are equal in length. It is known that the side length of the square is 12 feet.

2. According to the formula for the perimeter of a square, which states that the perimeter of a square is equal to the side length multiplied by 4, calculate the perimeter of the base square. Let the side length be s, then the perimeter p = 4s = 4 * 12 = 48 feet.

3. To find the lateral area of the prism, use the formula for the lateral area of a prism, which states that the lateral area is equal to the product of the perimeter of the base and the height of the prism.

4. The height of the prism given in the problem is 28 feet, therefore, the lateral area L.A. = perimeter * height = 48 * 28 = 1,344 square feet.

5. After the above calculations, the final lateral area is 1,344 square feet.

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>The side length of the square is 12 feet</note>.<note>All four sides are equal and the interior angles are all 90 degrees</note>.

Definition of Prism

In the problem diagram, <note>the base of the prism is two equal squares</note>, <note>the height of the prism is 28 feet</note>.

Lateral Surface Area Formula of a Prism

<note>The perimeter of the base of the prism is 48 feet</note>, <note>The height of the prism is 28 feet</note>. The lateral surface area is calculated as <note>L.A. = 48 * 28 = 1,344 square feet</note>.

Perimeter Formula for Square

The perimeter of a square is equal to four times the length of any one of its sides.

<note>The perimeter of a square is equal to four times the side length 12</note>, i.e., p = 12 * 4 = 48.


Question:
What is m $\angle $ W?

m $\angle $ W= $\Box$ °

Answer: m \$\angle \$ W=74°

1. Given that ∠TVU is an inscribed angle in the figure, with a measure of 37°.

2. The central angle ∠TWU and the inscribed angle ∠TVU subtend the same arc, which is arc TU.

3. According to the Inscribed Angle Theorem, in the same circle, the central angle subtending the same arc is twice the inscribed angle, thus ∠TWU = 2 × ∠TVU.

4. Substituting the measure of ∠TVU, we get ∠TWU = 2 × 37°.

5. Calculating, we get ∠TWU = 74°.

6. Through the above reasoning, the final answer is 74°.

Definition of Inscribed Angle

In the figure of this problem, in the circle, the vertex <note>V</note> of the angle <note>∠TVU</note> is on the circumference, and the two sides of the angle <note>∠TVU</note> intersect the circle at <note>point T and point U</note> respectively. Therefore, the angle <note>∠TVU</note> is an inscribed angle.

Definition of Central Angle

An angle formed by two radii connecting two points on the circumference of a circle to the center of the circle is called a central angle.

<note>Angle ∠TWU is a central angle</note>, with its vertex <note>W at the center of the circle</note>, and its two sides <note>WT and WU are both radii of the circle</note>, thus forming <note>angle TWU</note>.

Inscribed Angle Theorem

In the diagram of this problem, the <note>arc TU</note> corresponds to the <note>inscribed angle ∠TVU</note>, and the <note>arc TU corresponds to the central angle ∠TWU</note>. According to the Inscribed Angle Theorem, the <note>∠TVU</note> is equal to half of the central angle <note>∠TWU</note> that corresponds to the arc <note>TU</note>, that is, <note>∠TVU = 1/2 ∠TWU</note>. Therefore, <note>∠TWU = 2 × ∠TVU = 2 × 37° = 74°</note>.


Question:
Find the lateral area of the pyramid. The base is an equilateral triangle.

 $\Box$  square centimeters

Answer: 576 square centimeters

1. Given that the base is an equilateral triangle with a side length of 16 cm, according to the definition of an equilateral triangle, all sides are equal, so each side length is 16 cm.

2. According to the formula for the perimeter of a triangle, the perimeter of the triangle is the sum of all sides. For an equilateral triangle, the perimeter p = 3 × side length = 3 × 16 = 48 cm.

3. According to the formula for the lateral area of a regular pyramid, one of the lateral faces is a triangle whose base is the length of one side of the base, and the height is the slant height. The lateral area of a regular pyramid can be calculated using the formula L.A. = 1/2 × base perimeter × slant height.

4. The slant height given in the problem is 24 cm. Therefore, we can substitute into the formula to calculate the lateral area: L.A. = 1/2 × 48 × 24 = 576 square cm.

5. Through the above reasoning, the final answer is 576 square cm.

Formula for the Lateral Surface Area of a Regular Triangular Pyramid

The formula for the lateral surface area of a regular triangular pyramid is: \(S_{lateral}=(\frac{1}{2}) \times C \times h\), where \(C\) is the perimeter of the equilateral triangle base, and \(h\) is the height of any of the isosceles triangle faces.

<note>The perimeter of the equilateral triangle base C=16+16+16=48</note>, <note>The height of the isosceles triangular side is 24</note>, so the lateral surface area of the regular triangular pyramid <note>S_side=(1/2) * 48 * 24=576</note>.

Formula for the Perimeter of a Triangle

In the figure of this problem, in the triangle, <note>the three sides are 16, 16, 16 respectively</note>, according to <note>the formula for the perimeter of a triangle</note>, that is <note>perimeter L = 16 + 16 + 16 = 48</note>.


Question:
Find the area of △VWX.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
 $\Box$  mi ^ 2

Answer: 266.8 mi ^ 2

1. According to the problem, the lengths of the three sides of triangle △VWX are given: VW = 20 mi, WX = 30 mi, VX = 43 mi.

2. To calculate the area of the triangle, Heron's formula can be used. The formula is:

   Area = √[s(s - a)(s - b)(s - c)], where a, b, c are the three sides of the triangle,

   s is the semi-perimeter of the triangle, s = (a + b + c)/2.

3. Calculate the semi-perimeter s: s = (VW + WX + VX)/2 = (20 + 30 + 43)/2 = 93/2 = 46.5.

4. Calculate the area using Heron's formula:

   Area = √[46.5(46.5 - 20)(46.5 - 30)(46.5 - 43)].

5. Calculate each subtraction:

   46.5 - 20 = 26.5, 46.5 - 30 = 16.5, 46.5 - 43 = 3.5.

6. Calculate the product:

   46.5 × 26.5 × 16.5 × 3.5 = 71,162.4375.

7. The area is equal to the square root of the above product:

   Area = √71,162.4375 ≈ 266.7628.

8. Finally, according to the problem requirements, round the result to the nearest tenth:

   266.7628 rounded to the nearest tenth is 266.8.

9. After the above reasoning, the final answer is 266.8 square miles.

Definition of Triangle

A geometric figure is called a triangle if it is formed by three non-collinear points and the line segments connecting them.

In the problem diagram, the triangle <note>△VWX</note> is a geometric figure composed of <note>three non-collinear points V, W, X</note> and their connecting line segments <note>VW, WX, VX</note>. <note>Points V, W, X</note> are respectively the <note>three vertices</note> of the triangle, and the line segments <note>VW, WX, VX</note> are respectively the triangle's <note>three sides</note>.

Heron's Formula

Heron's formula is used to calculate the area of any triangle. The formula is given by: \( A = \sqrt{s(s - a)(s - b)(s - c)} \), where \( s \) is the semi-perimeter, and \( a, b, \) and \( c \) are the lengths of the sides of the triangle.

In the diagram of this problem, use Heron's formula to calculate the area of triangle <note>△VWX</note>. First, calculate the semi-perimeter <note>s</note>, which is <note>s = 46.5</note>. Then, substitute the lengths of the three sides into the formula: <note>VW = 20</note>, <note>WX = 30</note>, <note>VX = 43</note>. The area is calculated as <note>√[46.5(46.5 - 20)(46.5 - 30)(46.5 - 43)] ≈ 266.8 square miles</note>.


Question:
Find the lateral area of the cone.

Round your answer to the nearest tenth.

 $\Box$  square feet

Answer: 603.2 square feet

1. Given that the slant height of the cone is 24 feet and the radius of the base is 8 feet.

2. The lateral area (L.A.) of the cone can be calculated using the formula L.A. = πrℓ, where r is the radius of the base and ℓ is the slant height.

3. Substitute the given radius r = 8 feet and slant height ℓ = 24 feet into the lateral area formula, obtaining L.A. = π × 8 × 24.

4. Calculate the expression to get L.A. = 192π.

5. Using the approximation π ≈ 3.14159 for the calculation, we get L.A. ≈ 192 × 3.14159.

6. Continue the calculation to get approximately L.A. ≈ 603.1853.

7. Round the result to the nearest tenth as required by the problem, resulting in a lateral area of approximately 603.2 square feet.

8. Based on the above reasoning, the final answer is 603.2 square feet.

Generatrix

The generatrix of a cone is the line segment that joins a point on the circumference of the base to the apex.

In the figure of this problem, <note>The generatrix of the cone is 24 feet long</note>, represented as <note>the dashed line from the apex to any point on the base circumference</note>, with a length of 24 feet. <note>The generatrix is the line segment from a point on the base circumference to the apex in a cone</note>.

Lateral Surface Area of a Cone

Original: <note>底面半径 r = 8 英尺</note> 和 <note>母线长 ℓ = 24 英尺</note>，代入公式后得 L.A. = π × 8 × 24。
    Translation: <note>Base radius r = 8 feet</note> and <note>Slant height ℓ = 24 feet</note>, substituting into the formula gives L.A. = π × 8 × 24.


Question:
Find the lateral area of the cylinder.

Round your answer to the nearest tenth.

 $\Box$  square yards

Answer: 678.6 square yards

1. Given that the height of the cylinder is 12 yards and the diameter is 18 yards.

2. According to the formula for the lateral area of a cylinder in geometry, lateral area L.A. = 2πrh, where r is the radius of the base and h is the height of the cylinder.

3. Since the diameter of the base of the cylinder is 18 yards, its radius r = 18 ÷ 2 = 9 yards.

4. Substituting the radius of the base r = 9 yards and the height of the cylinder h = 12 yards into the lateral area formula, we get L.A. = 2π(9)(12).

5. Calculating, we get L.A. ≈ 2 × 3.14159 × 9 × 12.

6. After calculation, we get L.A. ≈ 678.583.

7. According to the problem requirements, the result is rounded to one decimal place, so the lateral area is approximately 678.6 square yards.

8. The final answer is 678.6.

Cylinder

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

<note>In the figure of this problem, the cylinder is composed of two parallel and identical circular bases and a lateral surface</note>. The bases are <note>two identical circles</note>, with <note>a diameter of 18 yards</note>, <note>a radius of 9 yards</note>, and <note>their centers are on the same line</note>. The lateral surface is <note>a rectangle</note>, when unfolded, its <note>height is equal to the height of the cylinder, 12 yards</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.

<note>The diameter of the circular base of the cylinder is 18 yards</note>, so its <note>radius r = 18 ÷ 2 = 9 yards</note>. The radius of a circle refers to <note>the length of the line segment from the center of the circle to any point on the circumference</note>.

Formula for Lateral Area of a Cylinder

Original text: <note>Radius of the cylinder r = 9 yards</note>, <note>Height of the cylinder h = 12 yards</note>, substituting into the formula to get the lateral area <note>L.A. = 2π(9)(12)</note>. After calculation, the lateral area is approximately 678.6 square yards.


Question:
Find the lateral area of the cylinder.

Round your answer to the nearest tenth.

 $\Box$  square meters

Answer: 2,412.7 square meters

1. Given that the height of the cylinder is 24 meters and the radius of the base circle is 16 meters.

2. The lateral surface area of the cylinder can be calculated using the formula for the lateral surface area of a cylinder: lateral surface area = 2πrh, where r represents the radius of the circle and h represents the height of the cylinder.

3. Substitute the given radius r = 16 meters and height h = 24 meters.

4. Lateral surface area = 2 × π × 16 × 24.

5. Use the standard approximation value for π, which is 3.141592653589793.

7. Round the calculation result to the nearest tenth.

9. Through the above reasoning, the final answer is 2412.7 square meters.

Cylinder

<note>In the figure of this problem, the cylinder consists of two parallel and identical circular bases and a lateral surface.</note> The bases are <note>two identical circles with a radius r = 16 meters</note>, and <note>their centers are on the same line</note>. The lateral surface is a rectangle, when unfolded, <note>its height h = 24 meters</note>, <note>and the width is equal to the circumference of the circle</note>.

Definition of Radius

In the figure of this problem, <note>the radius r of the base circle of the cylinder is 16 meters</note>, <note>the length of the line segment connecting the center of the circle to any point on the circumference is 16 meters</note>.

Formula for Lateral Area of a Cylinder

<note>The radius of the base circle of the cylinder r = 16 meters</note>, <note>height h = 24 meters</note>, the lateral area of the cylinder can be calculated using the formula <note>lateral area = 2πrh</note> = 2 × π × 16 × 24.


Question:
Find the lateral area of the cylinder.

Round your answer to the nearest tenth.

 $\Box$  square meters

Answer: 659.7 square meters

1. Given the height of the cylinder h = 15 meters, and the radius of the base r = 7 meters.

2. The formula for the Lateral Area (L.A.) of a cylinder is L.A.=2πrh, where r is the radius of the base and h is the height.

3. Substitute the given values r = 7 meters and h = 15 meters into the lateral area formula: L.A.=2π * 7 * 15.

4. Calculate 2π * 7 * 15, obtaining the value 210π.

6. Round the result 659.7344573 to one decimal place, obtaining 659.7.

7. Through the above reasoning, the final lateral area of the cylinder is 659.7 square meters.

Cylinder

<note>A cylinder consists of two parallel and identical circular bases and a lateral surface</note>. The bases are <note>two identical circles</note>, their radii and diameters are equal, and their centers lie on the same line. The lateral surface is a rectangle, and when unfolded, <note>its height is equal to the height of the cylinder</note>, <note>its width is equal to the circumference of the circle</note>. Specifically, the cylinder's <note>two circular bases have radii of r = 7 meters</note>, <note>height is h = 15 meters</note>.

Definition of Radius

In the diagram of this problem, the point in the base circle of the cylinder is the center of the circle, any point on the circumference, the line segment is the line segment from the center of the circle to any point on the circumference, therefore the line segment of 7 meters is the radius of the circle.

Formula for Lateral Area of a Cylinder

In the figure of this problem, it is known that <note>the radius of the base of the cylinder r = 7 meters</note> and <note>the height h = 15 meters</note>. Substitute the known r and h into the formula for lateral area to get <note>L.A.=2π* 7 * 15</note>. After calculation, the lateral area is <note>210π</note>. Using the approximate value <note>π ≈ 3.141592653</note>, further calculation gives <note>210π≈659.7344573</note>, rounding the result to one decimal place gives the final answer <note>659.7 square meters</note>.


Question:
Find the lateral area of the cone.

Round your answer to the nearest tenth.

 $\Box$  square yards

Answer: 351.9 square yards

1. Given that the slant height (length of the generatrix) of the cone is 14 yards, and the radius of the base is 8 yards.

2. The formula for calculating the lateral surface area of the cone is: lateral surface area = π × radius × slant height.

3. Substitute the known values into the formula: lateral surface area = π × 8 × 14.

4. Calculate the result: lateral surface area = 112π.

5. Use a calculator to find the approximate value of 112π: 351.85837720205683.

6. Round the calculated result to the nearest tenth: lateral surface area ≈ 351.9 square yards.

7. Through the above reasoning, the final answer is 351.9 square yards.

Definition of Cone

<note>The radius of the base is 8 yards</note>, <note>The slant height of the generatrix is 14 yards</note>, these elements determine the shape of the cone.

Generatrix

<note>The slant height of the cone is 14 yards</note>, that is, <note>the distance from the vertex of the cone to any point on the circumference of the base is 14 yards</note>. <note>The generatrix is the line segment from a point on the circumference of the base to the vertex in the cone</note>.

Lateral Surface Area of a Cone

In the figure of this problem, according to the given <note>base radius of 8 yards</note> and <note>slant height of 14 yards</note>, the lateral surface area calculation formula is <note>lateral surface area = 𝜋 × 8 × 14</note>. Substituting the known values into the formula, the calculated result is <note>lateral surface area = 112𝜋</note>.


Question:
Find the lateral area of the cone.

Round your answer to the nearest tenth.

 $\Box$  square meters

Answer: 37.7 square meters

1. Given that the slant height of the cone is 6 meters, and the diameter of the base is 4 meters.

2. The diameter of the base is known, so the radius r of the base is half of the diameter, i.e., r = 4 / 2 = 2 meters.

3. The formula for the Lateral Area (L.A.) of the cone is L.A = π * r * 𝓁, where r is the radius of the base and 𝓁 is the slant height.

4. Substitute the known values into the formula: L.A = π * 2 * 6 = 12π.

6. According to the problem requirements, round the result to one decimal place: 37.7.

7. Through the above reasoning, the final answer is 37.7 square meters.

Definition of Cone

The length of the <note>generatrix (slant height) of the cone</note> is 6 meters, the <note>diameter of the base</note> is 4 meters, so the <note>radius r</note> is 2 meters.

Definition of Radius

In the figure of this problem, <note>the diameter of the cone's base is 4 meters</note>, so <note>the radius of the base r is half of the diameter, r = 2 meters</note>. <note>The radius of a circle is the length of the line segment from the center of the circle to any point on the circumference</note>.

Lateral Surface Area of a Cone

The original text: <note>底面半径r为2米</note>, <note>母线长度𝓁为6米</note>, according to the formula, the lateral surface area can be calculated: L.A = π * 2 * 6 = 12π.


Question:
Look at this figure:What is the shape of its base?

- rectangle
- circle
- square
- pentagon

Answer: - circle

1. It is known that this geometric figure is a cone.

2. According to the definition of a cone, a cone is a geometric figure with a circular base and an axis from the base to the vertex that is not in the same plane.

3. From this, it can be concluded that the base of this cone is a circle.

4. Through the above reasoning, the final answer is a circle.

Definition of Cone

In a cone, the generatrix rotates around the axis to form the <note>conical surface and base (circle)</note>, thereby forming a cone.

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 base of the cone is circular. The distance from every point on the base of the cone to the <note>center of the circle</note> is equal, which conforms to the definition of a circle. The distance from all points on the base of the cone to the center of the circle is equal to the <note>radius</note>.