Question: The positions of lines a, b, c, and d are as shown in the figure. If ∠1=58°, ∠2=58°, ∠3=70°, then ∠4 equals ()

Answer: 110°

1. Given ∠1 = 58°, ∠2 = 58°, according to the definition of corresponding angles, if two lines are cut by a third line and the corresponding angles are equal, then the two lines are parallel. Therefore, line a is parallel to line b.

2. Let the opposite angle of ∠4 be ∠5, according to the definition of vertical angles, ∠4 = ∠5.

3. According to the problem, ∠3 = 70°, since line a is parallel to line b, based on the parallel postulate 2, the interior angles on the same side are supplementary, then ∠5 + ∠3 = 180°, thus ∠5 + 70° = 180°, i.e., ∠5 = 180° - 70° = 110°.

4. Since step 2 gives ∠4 = ∠5, therefore, ∠4 = ∠5 = 110°.

5. Through the above reasoning, the final answer is 110°.

平行线; 内错角; 同位角

Parallel Postulate 2 of Parallel Lines

In the diagram of this problem, <note>line a and line b are intersected by line d</note>, forming the following geometric relationships: 1. <note>Corresponding angles: ∠1 and ∠2 are equal</note>. 2. <note>Consecutive interior angles: ∠3 and ∠5 are supplementary, i.e., ∠3 + ∠5 = 180 degrees</note>. These relationships illustrate that <note>when two parallel lines are intersected by a third line</note>, <note>corresponding angles are equal</note>, <note>consecutive interior angles are supplementary</note>.

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.

In the figure of this problem, two lines <note>a and b</note> are intersected by a third line <note>d</note>. The two angles <note>3 and 5 are on the same side of the intersecting line d and within the intersected lines a and b</note>, so <note>angles 3 and 5 are consecutive interior angles</note>. Consecutive interior angles <note>3 and 5 are supplementary</note>, that is, <note>angle 3 + angle 5 = 180 degrees</note>.

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>Two parallel lines a and b</note> are intersected by <note>a line c</note>, where <note>angle 1 and angle 2 are on the same side of the intersecting line c and on the same side of the two intersected lines a and b</note>, thus <note>angle 1 and angle 2 are corresponding angles</note>. <note>Corresponding angles are equal, that is, angle 1 is equal to angle 2</note>.

Definition of Vertical Angles

In the diagram for this problem, two intersecting lines <note>b and d intersect at a point</note>, forming <note>two angles: angle 4 and angle 5</note>. According to the definition of vertical angles, <note>angle 4 and angle 5 are vertical angles</note>. Since vertical angles are equal, <note>angle 4 = angle 5</note>.


Question: If ABCD is a rhombus, and m \angle ABC = 70, find m \angle 1?

Answer: 55

1. Given that quadrilateral ABCD is a rhombus, according to the properties of a rhombus, the diagonals of a rhombus bisect each other and are perpendicular.

2. Given ∠ABC = 70°, and given BA = BC, according to the definition of an isosceles triangle, triangle ABC is an isosceles triangle, therefore ∠1 = ∠ACB.

3. In triangle ABC, given ∠ABC = 70°, according to the triangle sum theorem, we can deduce that 2∠1 + ∠ABC = 180°.

####

4. Through the above reasoning, the final answer is 55°.

菱形; 等腰三角形; 三角形的外角

Definition of Rhombus

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

<note>In the figure of this problem, in quadrilateral ABCD, all sides AB, BC, CD, and DA are equal</note>, so quadrilateral ABCD is a rhombus. <note>Additionally, the diagonals AC and BD of quadrilateral ABCD are perpendicular bisectors of each other, meaning diagonals AC and BD intersect at point O</note>, and <note>∠AOB = ∠BOC = ∠COD = ∠DOA = 90°</note>, and <note>OA=OB=OC=OD</note>.

Definition of an Angle

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

<note>Angle ∠ABC is a geometric figure formed by rays BA and BC</note>, <note>these two rays share a common endpoint B</note>. <note>This common endpoint B is called the vertex of angle ∠ABC</note>, and <note>rays BA and BC are called the sides of angle ∠ABC</note>.

Triangle Angle Sum Theorem

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

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

Definition of Isosceles Triangle

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

Original: <note>In triangle ABC</note>, <note>side BA and side BC are equal</note>, therefore <note>triangle BAC is an isosceles triangle</note>.

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.

<note>In the figure of this problem, in the isosceles triangle ABC, side BA and side BC are equal</note>. Therefore, according to the properties of an isosceles triangle, the angles opposite the equal sides are equal, that is, <note>angle BAC = angle BCA</note>.


Question: (4 points) As shown in the figure, line a ∥ b, line c intersects a and b respectively, ∠1 = 50°, then the degree of ∠2 is ()

Answer: 130°

1. ##Let line c intersect line a at point O##.

2. ##According to the definition of vertical angles, ∠1 and ∠DOa are vertical angles and have equal measures, ∠DOa=∠1=50°. Since line a∥b, according to the definition of same-side interior angles, ∠2 and ∠DOa are same-side interior angles. According to the parallel postulate 2, same-side interior angles are supplementary, so ∠2+∠DOa=180°##.

3. ##Substituting the values into the equation gives ∠2+50°=180°##.

4. ##Calculating gives ∠2=180°-50°=130°##.

####

##5##. After the above reasoning, the final answer is 130°.

Definition of Consecutive Interior Angles

<note>Two lines a and b</note> are cut by a third <note>line c</note>, with two angles <note>∠2</note> and <note>∠DOa</note> on the same side of the transversal <note>c</note> and inside the lines being cut <note>a and b</note>, so <note>∠2</note> and <note>∠DOa</note> are consecutive interior angles.

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 a and c intersect at point O</note>, forming angles: <note>∠DOa, ∠1</note>. According to the definition of vertical angles, <note>∠1 and ∠DOa are vertical angles</note>, and since vertical angles are equal, <note>∠DOa = ∠1</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.

<note>Line a is parallel to line b</note>, <note>line c intersects lines a and b at points O and D respectively</note>, forming a geometric relationship: same-side interior angles, <note>∠2 and ∠DOa are supplementary</note>, that is, <note>∠2 + ∠DOa = 180°</note>.


Question: Use the parallelogram to find b.

Answer: 11

1. Given that quadrilateral XYZW is a parallelogram. According to the properties of a parallelogram ##theorem##, which states that the opposite sides of a parallelogram are equal, we have WX = YZ and XY = WZ.

2. ##Given## WX = YZ, ##from the figure## we get: b + 11 = 2b.

3. Solving the equation b + 11 = 2b, we get b = 11.

4. Through the above calculations and the properties of the parallelogram, we finally obtain the value of b as 11.

Definition of Parallelogram

A quadrilateral is a parallelogram if and only if it has two pairs of opposite sides that are both parallel (∥) and congruent (≅).

<note>Quadrilateral XYZW is a parallelogram</note>, <note>side XY is parallel and equal to side WZ</note>, <note>side WX is parallel and equal to side YZ</note>.

Properties of Parallelogram Theorem

In a parallelogram, opposite angles are equal, opposite sides are equal, and the diagonals bisect each other.

In the <note>parallelogram XYZW</note>, <note>the opposite angles ∠X and ∠Z are equal</note>, <note>the opposite angles ∠W and ∠Y are equal</note>; <note>the sides WX and YZ are equal</note>, <note>the sides XY and WZ are equal</note>.


Question: As shown in the figure, C is a point on ⊙O, O is the center of the circle, if ∠C=35°, then the degree of ∠AOB is ()

Answer: 70°

1. Let O be the center of the circle, and C be any point on the circumference.

2. According to the definition of the inscribed angle, ∠C is an inscribed angle and equals 35°.

3. According to the definition of the central angle, ∠AOB is a central angle. According to the inscribed angle theorem, ∠C = ½ ∠AOB.

4. Calculation yields ∠AOB = 2 ∠C = 2 × 35° = 70°.

5. Through the above reasoning, the final answer is ∠AOB = 70°.

圆; 圆周角; 圆心角; 弧; 三角形的外角

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.

<note>In circle O, the vertex C of angle ACB is on the circumference</note>, <note>the two sides of angle ACB intersect circle O at points A and B respectively</note>. Therefore, <note>angle ACB 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 and point B are two points on the circle</note>, <note>the center of the circle is point O</note>. The angle ∠AOB formed by <note>line segments OA and OB</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.

<note>In the figure of this problem</note>, in circle O, <note>points A, B, and C are on the circle</note>, the central angle corresponding to <note>arc AB</note> is ∠AOB, and the inscribed angle is ∠ACB. According to the Inscribed Angle Theorem, ∠ACB is equal to half of the central angle ∠AOB corresponding to <note>arc AB</note>, that is, <note>∠ACB = ½∠AOB</note>, then ∠AOB=<note>2∠ACB=2∠C</note>.


Question: Find K L.

Answer: 12

1. Given that both triangle XYZ and triangle JKL have an angle of 51°, ##and angle X = angle J##.

2. ##According to the theorem of similar triangles (AA), triangle XYZ is similar to triangle JKL.##

3. ##According to the definition of similar triangles, corresponding sides are proportional, i.e., XY/JK = YZ/KL.##

4. ##Substitute the given side lengths.##

5. ##Obtain 5/4 = 15/x.##

6. We can ##solve the equation## to find x:

   x = 12.

7. In conclusion, KL = 12.

普通三角形; 等腰三角形

Similarity Theorem for Triangles (AA)

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

<note>Angle X is equal to angle J</note>, and <note>angle Y is equal to angle K</note>, so <note>triangle XYZ is similar to triangle JKL</note>.

Definition of Similar Triangles

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

<note>Triangle XYZ and triangle JKL are similar triangles</note>. According to the <note>definition of similar triangles</note>, we have: <note>angle X = angle J</note>, <note>angle Y = angle K</note>, <note>angle Z = angle L</note>; <note>YZ/KL = XZ/JL = XY/JK</note>.


Question: In the figure shown above, AC = 6. What is the length of segment AB?

Answer: 6

1. Given that in △ABC, ∠BAC = 60°, ∠ABC = 60°, according to the triangle angle sum theorem, ∠ACB + ∠BAC + ∠ABC = 180°, given ∠BAC = 60°, ∠ABC = 60°, so ∠ACB = 180° - ∠BAC - ∠ABC = 180° - 60° - 60° = 60°. According to the angle properties of an equilateral triangle, △ABC is an equilateral triangle.

2. According to the definition of an equilateral triangle, in an equilateral triangle, the lengths of the three sides are equal. Specifically applied here: AB = AC = BC.

3. Given AC = 6, so AB = 6.

4. Through the above reasoning process, it is concluded that all sides of △ABC are of length 6, including AB.

Triangle Angle Sum Theorem

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

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

Angle Property of Equilateral Triangle

Each interior angle of an equilateral triangle is 60°.

In the figure of this problem, <note>in the equilateral triangle ABC, sides AB, BC, and AC are equal</note>, therefore, according to the properties of an equilateral triangle, <note>each interior angle of triangle ABC is 60°</note>. That is to say, <note>angle ABC, angle BAC, and angle ACB are all 60°</note>.

Definition of Equilateral Triangle

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

<note>Triangle ABC is an equilateral triangle</note>. <note>The lengths of sides AB, BC, and AC are equal</note>, and <note>the degrees of angles BAC, ABC, and ACB are equal, each being 60°</note>. Given <note>AC = 6</note>, therefore <note>AB = 6</note>.


Question: Find x.

Answer: 120

1. Given that the three interior angles of the triangle are equal. According to the triangle interior angle sum theorem, each interior angle is equal to 60 degrees, therefore, this is an equilateral triangle.

2. According to the triangle exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. As shown in the figure, the x° angle is an exterior angle of the triangle, equal to the sum of the other two non-adjacent interior angles. Given that each angle of an equilateral triangle is 60 degrees, so x°=60°+60°=120°.

3. After the above reasoning, the final answer is 120 degrees.

普通三角形; 三角形的外角; 直线

In the figure of this problem, the triangle is <note>an equilateral triangle</note>. <note>The lengths of the three sides are equal</note>, and <note>the measures of the three interior angles are equal, each being 60°</note>.

Angle Property of Equilateral Triangle

Original text: <note>In an equilateral triangle, all three sides are equal</note>, therefore according to the properties of an equilateral triangle, <note>each interior angle of the triangle is 60°</note>. This means that each angle is 60°.

Exterior Angle Theorem of Triangle

An exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

<note>The interior angles of the triangle are ∠1, ∠2, ∠3</note>. In the triangle, <note>x° angle is an exterior angle of the triangle</note>, <note>∠1 and ∠2 are the two interior angles that are not adjacent to the exterior angle x°</note>. According to the Exterior Angle Theorem of Triangle, the exterior angle is equal to the sum of the two non-adjacent interior angles, that is, <note>x° angle = ∠1 + ∠2</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.

In the figure of this problem, one interior angle of the triangle is the right base angle, and the <note>angle X</note> formed by extending the adjacent side and the other side of this angle is called the exterior angle of this interior angle.

Triangle Angle Sum Theorem

<note>In the diagram of this problem, the sum of the three interior angles of a triangle is equal to 180°</note>.


Question: m \angle 4 = 42. Find m \angle 7.

Answer: 138

1. Given line a ∥ line b, suppose line a and line b are intersected by line c.

2. According to the definition of adjacent supplementary angles, ∠3 and ∠4 are adjacent supplementary angles, given ∠4 = 42°, ∠3 = 180° - ∠4 = 180° - 42° = 138°.

3. According to Parallel Postulate 2 and the definition of corresponding angles, ∠3 = ∠7.

4. Therefore, we get m∠7 = 138°.

5. Through the above reasoning, the final answer is m∠7 = 138°.

Definition of Line

A line is a one-dimensional geometric element that has no width or thickness and extends infinitely in both directions.

In the figure of this problem, <note>line a</note> and <note>line b</note> are one-dimensional geometric elements without width and thickness. They extend infinitely in both directions, with no endpoints or boundaries.

Definition of Parallel Lines

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

<note>Line a and line b lie in the same plane</note>, and <note>they do not intersect</note>, so according to the definition of parallel lines, <note>line a and line b are parallel lines</note>.

Definition of Linear Pair of Angles

Two angles are in a linear pair if they share a common side, and their other sides are extensions of one another in opposite directions.

Angle 4 and angle 3 have a common side, their other sides are extensions in opposite directions, so angle 3 and angle 4 are adjacent supplementary angles.

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.

In the figure of this problem, <note>two parallel lines a and b are intersected by a line c</note>, where <note>angle 3 and angle 4 are on the same side of the intersecting line c, on the same side of the two intersected lines a and b</note>, thus <note>angle 3 and angle 7 are corresponding angles</note>. <note>Corresponding angles are equal</note>, that is, <note>angle 3 is equal to angle 7</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 diagram of this problem, two parallel lines <note>a and b</note> are intersected by a third line <note>c</note>, forming the following geometric relationships:
1. <note>Corresponding angles: angle 3 and angle 7 are equal</note>.
2. <note>Alternate interior angles: angle 4 and angle 6 are equal</note>.
3. <note>Consecutive interior angles: angle 4 and angle 5 are supplementary, i.e., angle 4 + angle 5 = 180 degrees</note>.
These relationships illustrate that when two parallel lines are intersected by a third line, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.##


Question: As shown in the figure, to measure the height of the school's flagpole, Xiao Dong uses a bamboo pole of length 3.2 meters as a measuring tool. He moves the bamboo pole so that the top end of the bamboo pole and the top end of the flagpole's shadow fall exactly at the same point on the ground. At this moment, the bamboo pole is 8 meters away from this point and 22 meters away from the flagpole. Then the height of the flagpole is () meters.

Answer: 12

1. Let the height of the bamboo pole be BC=3.2 (letters ordered from top to bottom), the height of the flagpole be EF=x (letters ordered from top to bottom), and the point where the shadows of the top of the bamboo pole and the top of the flagpole just fall on the ground be point A.

2. Since the tops of the shadows of the bamboo pole and the flagpole coincide at the same point, △ABC and △AEF share ∠A. Because BC and EF are the heights of the bamboo pole and flagpole respectively, according to the definition of height, BC⊥AE, EF⊥AE, so ∠BCA=∠EFA=90°. According to the similarity theorem (AA), △ABC∽△AEF.

3. From the problem statement, AF=AC + CF = 8 + 22 = 30.

4. According to the definition of similar triangles, we get the relationship: AC/AF = BC/EF.

5. Substitute the known lengths into the proportion relationship: 8 / 30=3.2 / x.

6. Solve the equation: x = 3.2 * (30 / 8).

8. Through the above reasoning, the final answer is 12 meters.

普通三角形; 直角三角形

Definition of Similar Triangles

In the figure of this problem, <note>△ABC and △AEF are similar triangles</note>. According to the definition of similar triangles: <note>AC/AF = BC/EF = AB/AE</note>.

Similarity Theorem for Triangles (AA)

In the figure of this problem, let the height of the bamboo pole be <note>BC=3.2 (letters ordered from top to bottom)</note>, and the height of the flagpole be <note>EF=x (letters ordered from top to bottom)</note>. The shadow of the top of the bamboo pole and the top of the flagpole falls exactly at the same point on the ground, designated as point A. In △<note>ABC</note> and △<note>AEF</note>, because <note>∠A=∠A</note> and <note>∠BCA=∠EFA=90°</note>, therefore △<note>ABC is similar to △AEF</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).

<note>Assume the height of the flagpole is EF</note>. The line segment <note>EF from vertex E perpendicular to the opposite side AF</note> is the altitude from <note>vertex E</note>. The line segment <note>EF forms a right angle (90 degrees) with side AF</note>, indicating that the line segment <note>EF is the perpendicular distance from vertex E to the opposite side AF (or its extension)</note>, so <note>∠EFA=90°</note>. Similarly, assume the height of the bamboo pole is <note>BC</note>. The line segment <note>BC from vertex B perpendicular to the opposite side AF</note> is the altitude from <note>vertex B</note>. The line segment <note>BC forms a right angle (90 degrees) with side AF</note>, indicating that the line segment <note>BC is the perpendicular distance from vertex B to the opposite side AF (or its extension)</note>, so <note>∠BCA=90°</note>. Therefore, <note>∠BCA=∠EFA=90°</note>.


Question: As shown in the figure, D is the intersection point of the angle bisectors BD and CD of △ABC. If ∠A=50°, then ∠D=()

Answer: 115°

1. Given angle A is equal to 50 degrees, according to the triangle angle sum theorem, angle ABC plus angle ACB equals 180 degrees minus angle A, that is, 180 degrees minus 50 degrees equals 130 degrees.

2. From the above conclusion, and D is the intersection point of the angle bisectors BD and CD of triangle ABC, according to the ##definition of angle bisector##, angle DBC plus angle DCB equals 1/2 times (angle ABC plus angle ACB), that is, 1/2 times 130 degrees equals 65 degrees.

3. In triangle BCD, according to the triangle angle sum theorem, angle D equals 180 degrees minus (angle DBC plus angle DCB), that is, 180 degrees minus 65 degrees equals 115 degrees.

4. Through the above reasoning, the final answer is angle D equals 115 degrees.

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

Triangle Angle Sum Theorem

<note>In the figure of this problem, in triangle ABC, angle A, angle B, and angle C are the three interior angles of triangle ABC</note>, according to the Triangle Angle Sum Theorem, <note>∠A + ∠B + ∠C = 180°</note>.
<note>In triangle BCD, angle B, angle C, and angle D are the three interior angles of triangle BCD</note>, according to the Triangle Angle Sum Theorem, <note>∠B + ∠C + ∠D = 180°</note>.


Question: Find \sin S.

Answer: 0.6

1. Given triangle △ STR, ∠ STR is a right angle (90 degrees), TS = 4, TR = 3, SR = 5. ##According to the definition of a right triangle, △ STR is a right triangle##.

2. According to ##the definition of the sine function, in Rt△ STR, the opposite side of ∠ S is TR, and the hypotenuse is SR, so sin∠ S=TR/SR##.

####

##3##. Substituting the specific side lengths, we get ##sin ∠ S = TR/SR = 3/5=0.6##.

Definition of Right Triangle

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

In the figure of this problem, in triangle STR, <note>angle STR is a right angle (90 degrees)</note>, so triangle STR is a right triangle. <note>Side TS and side TR are the legs, and side SR is the hypotenuse</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 <note>right triangle STR</note>, <note>angle TSR is an acute angle</note>, <note>side TR is the opposite side of angle TSR</note>, and <note>side SR is the hypotenuse</note>. According to the definition of the sine function, the <note>sine value of angle TSR</note> is equal to the <note>ratio of the opposite side TR to the hypotenuse SR</note>, i.e., <note>sin(∠TSR) = TR / SR</note>.


Question: Find the area of the regular polygon. Round the result to the nearest tenth.

Answer: 65

1. Given that the distance from the center of a regular hexagon to any of its vertices is 5 meters, ##according to the definition of a regular polygon##, all its sides are equal in length and its internal angles are equal.

2. ##Let the center of the regular hexagon be O, then the distance from the six vertices to the center O is 5 meters. The line segments from the six vertices to the center O can divide the regular hexagon into 6 triangles. According to the congruent triangles theorem (SSS), since the distance from the six vertices to the center O is 5 meters and the six sides of the regular hexagon are equal, these six triangles are congruent.##

3. ##From the figure, it can be seen that the sum of the six angles at the center O is 360°, so one angle is 360°/6 = 60°, resulting in six congruent triangles being congruent equilateral triangles with a side length of 5 meters.##

4. ##The area of a regular polygon = the area of an equilateral triangle * 6. According to the triangle area formula (using the sine function), the area of an equilateral triangle = (side length^2 * √3) / 4.##

5. Substituting the side length of 5 meters into the formula, we get: area = (5^2 * √3) / 4 = (25 * √3) / 4 ≈ 10.8253 square meters.

####

##6. Total area = 10.8253 * 6 ≈ 64.9518 square meters.##

##7##. Finally, through the above reasoning, the answer is 64.9518 square meters. Rounding to one decimal place, the result is 65.0 square meters.

正多边形; 旋转; 正弦; 余弦; 线段

Definition of Regular Polygon

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

In the figure of this problem, <note>in a regular hexagon, all sides are of equal length</note>, and <note>all interior angles are equal</note>. Therefore, a regular hexagon is a regular polygon. <note>The distance from the center of the regular hexagon to any vertex is 5 meters</note>.

Definition of Equilateral Triangle

In the figure of this problem, the regular hexagon can be divided into 6 congruent equilateral triangles. <note>Each equilateral triangle has three sides of equal length, each being 5 meters</note>, and <note>each interior angle is equal, each being 60°</note>.

Triangle Congruence Theorem (SSS)

Two triangles are congruent if their three pairs of corresponding sides are equal in length.

<note>The line segments from the six vertices to the center O are 5 meters</note>, <note>In a regular hexagon, all sides are of equal length</note>, therefore according to the Triangle Congruence Theorem (SSS), <note>The six triangles in the figure are congruent</note>.

Triangle Area Formula (Using Sine Function)

The area \( S \) of any triangle can be expressed as \( S = \frac{1}{2} \cdot a \cdot b \cdot \sin(C) \), where \( a \) and \( b \) are the lengths of two sides, and \( C \) is the angle between these two sides.

In the figure of this problem, the six triangles are equilateral and congruent triangles, <note>each side length is 5 meters</note>, <note>and each angle is 60°</note>. According to the triangle area formula, <note>the area S of one equilateral triangle in the problem can be expressed as S = (1/2) * a * b * sin(C)</note>, that is, <note>S = (1/2) * 5 * 5 * sin(60°)</note>.


Question: As shown in the figure, when planting trees on flat ground, the tree spacing (the horizontal distance between two adjacent trees) is required to be 4m. If trees are planted on a slope with a gradient of 0.75, and the tree spacing is also required to be 4m, then the slope distance between two adjacent trees is ()

Answer: 5m

1. Given the slope is 0.75 and the planting distance, which is the horizontal distance, is 4 meters, let the slope angle be ∠BAC. Draw the auxiliary line AB as the horizontal distance, and CB perpendicular to AB, then ∠B=90°. According to the definition of a right triangle, triangle ABC is a right triangle.

2. Since the slope is 0.75, that is, tan(∠BAC)=0.75, according to the definition of the tangent function, tan(∠BAC)=CB/AB.

3. Since AB=4 meters and tan(∠BAC)=0.75, from the above equation we get: CB = 0.75 × AB = 0.75 × 4 = 3 meters.

4. According to the Pythagorean theorem, in the right triangle ABC, the length of AC (i.e., the slope distance) can be expressed as: AC=√(AB^2+CB^2).

5. Substituting the known lengths of AB and CB, we get AC=√(4^2+3^2)=√(16+9)=√25=5 meters.

6. Through the above reasoning, the final answer is 5 meters.

直角三角形; 正弦; 余弦

Definition of Right Triangle

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

Definition of Tangent Function

In a right-angled triangle, the tangent value of an acute angle is equal to the length of the opposite side divided by the length of the adjacent side.

In the <note>right triangle ABC</note>, <note>∠BAC is an acute angle</note>, <note>side CB is the side opposite to ∠BAC</note>, <note>side AB is the side adjacent to ∠BAC</note>, so <note>the tangent of ∠BAC is equal to the length of side CB divided by the length of side AB</note>, i.e., <note>tan(∠BAC) = CB / AB</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.

<note>In the right triangle ABC</note>, <note>∠ABC is a right angle (90 degrees)</note>, the sides <note>AB and CB are the legs</note>, the side <note>AC is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>AC^2 = AB^2 + CB^2</note>.


Question: The area of trapezoid J K L M is 138 square feet. The area of trapezoid Q R S T is 5.52 square feet. If trapezoid J K L M \sim trapezoid Q R S T, find the value of x.

Answer: 1

1. Given trapezoid JKLM and trapezoid QRST are similar, according to the ##theorem of the area ratio of similar polygons equals the square of the similarity ratio##, let the similarity ratio of trapezoid JKLM and trapezoid QRST be k.

2. According to the theorem that the area ratio equals the square of the similarity ratio, we get the area ratio relationship 138 / 5.52 = k^2.

3. Calculate the ratio 138 / 5.52 = 25, obtaining k^2 = 25.

4. Solve for the positive value of the similarity ratio k, obtaining k = 5.

5. Since trapezoid JKLM and trapezoid QRST are similar, corresponding sides are proportional, so the side JK of trapezoid JKLM and the side QT of trapezoid QTRS are in a 5:1 ratio due to the corresponding 5.

6. Therefore, x = 5 / 5 = 1.

7. Finally, the value of x is 1.

Definition of Similar Figures

Two geometric figures are similar if and only if their corresponding sides are proportional, and their corresponding angles are equal.

<note>Figures JKLM and QTRS are similar figures</note>. According to the definition of similar figures, the ratios of their corresponding sides are equal, that is, <note>the ratio of side JK to side QT is equal to the ratio of side ML to side RS</note>, and it is equal to <note>the ratio of side QR to side LK</note>. Additionally, their corresponding angles are also equal, that is, <note>angle MLK is equal to angle SRQ</note>, <note>angle TQR is equal to angle JKL</note>.

Definition of Trapezoid

A quadrilateral is a trapezoid if and only if it has exactly one pair of parallel sides.

<note>Side JK and side ML are parallel</note>, while <note>side JM and side KL are not parallel</note>. Therefore, according to the definition of a trapezoid, quadrilateral JKLM is a trapezoid because it has exactly one pair of parallel sides. In quadrilateral QRST, <note>side QT and side RS are parallel</note>, while <note>side QR and side ST are not parallel</note>. Therefore, according to the definition of a trapezoid, quadrilateral QRST is a trapezoid because it has exactly one pair of parallel sides.

Area Ratio Theorem of Similar Polygons

If the ratio of the side lengths of two similar polygons is k, then the ratio of their areas is equal to k squared (k²).

<note>Polygon JKML and polygon QTRS are similar figures</note>, <note>the ratio of side JK to side QT is k</note>, which means the similarity ratio is k. Therefore, <note>the ratio of the area of figure JKML to the area of figure QTRS is equal to the square of k</note>. That is: <note>area ratio = k?</note>.


Question: As shown in the figure, a // b, place the right-angle vertex of a triangular plate on line a, ∠1 = 42°, then the degree of ∠2 is ()

Answer: 48°

1. Given ∠1=42°, ##because the right-angle vertex of the triangle board is on line a, i.e., ∠CAB=90°, ∠3=∠CAB-∠1=90°-42°=48°##.

2. Since line a is parallel to line b, according to ##Parallel Postulate 2 and the definition of alternate interior angles, ∠3=∠2, thus ∠2=48°##.

3. Through the above reasoning, ##finally we conclude that the measure of ∠2 is 48°##.

平行线; 同位角; 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>Triangle ABC</note> is a geometric figure composed of <note>three non-collinear points A, B, C</note> and their connecting <note>line segments AC, AB, BC</note>. <note>Points A, B, C</note> are respectively <note>the three vertices of the triangle</note>, and <note>line segments AC, AB, BC</note> are respectively <note>the three sides of the triangle</note>.

Definition of Right Triangle

<note>Angle ∠CAB is a right angle (90 degrees)</note>, therefore triangle ABC is a right triangle. <note>Side AC and side AB are the legs, side BC is the hypotenuse</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 diagram of this problem, two parallel lines a and b are intersected by a line AB</note>, where <note>angle 3 and angle 2 are located between the two parallel lines and on opposite sides of the intersecting line AB</note>, therefore <note>angle 2 and angle 3 are alternate interior angles</note>. <note>Alternate interior angles are equal, that is, angle 2 is equal to angle 3</note>.

Parallel Postulate 2 of Parallel Lines

In the diagram of this problem, two parallel lines <note>a and b</note> are intersected by a third line <note>AB</note>, forming the following geometric relationships: 1. <note>Corresponding angles are equal</note>. 2. <note>Alternate interior angles: angle 2 and angle 3 are equal</note>. 3. <note>Same-side interior angles are supplementary</note>. These relationships illustrate that when two parallel lines are intersected by a third line, <note>corresponding angles are equal</note>, <note>alternate interior angles are equal</note>, <note>same-side interior angles are supplementary</note>.


Question: As shown in the figure, △ABC is an inscribed triangle of ⊙O. If ∠ABC=70°, then the degree of ∠AOC is equal to ()

Answer: 140°

1. With point O as the center and any length as the radius, draw a circle, referred to as ⊙O.

2. Select three different points A, B, and C on the circumference, such that △ABC passing through A, B, and C is an inscribed triangle of ⊙O.

3. Given ∠ABC = 70°.

4. According to the theorem of the inscribed angle and central angle, there is a relationship between the inscribed angle and central angle subtended by the same arc in a circle or equal arcs of the same circle: the measure of the central angle is twice that of the inscribed angle subtended by the arc.

5. In this problem, ∠ABC is the inscribed angle subtended by arc AC, and ∠AOC is the central angle subtended by arc AC.

6. Applying the above theorem, we can deduce: ∠AOC = 2 × ∠ABC.

7. Substituting the given ∠ABC = 70° into the above formula, we get: ∠AOC = 2 × 70° = 140°.

8. Through the above reasoning, the final answer is 140°.

圆周角; 圆心角; 圆; 普通三角形; 弧

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

Inscribed Angle Theorem

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

<note>In the figure of this problem, in circle O, points A, B, and C are on the circle</note>, <note>the arc AC and the central angle corresponding to arc AC is ∠AOC</note>, <note>the inscribed angle is ∠ABC</note>. According to the Inscribed Angle Theorem, <note>∠ABC is equal to half of the central angle ∠AOC corresponding to the arc AC</note>, that is, <note>∠ABC = 1/2 ∠AOC</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 the figure of this problem, in circle O, <note>point A and point C are two points on the circle</note>, <note>the center of the circle is point O</note>. <note>The angle ∠AOC formed by the lines OA and OC is called the central angle</note>.


Question: As shown in the figure, AB∥CD, ∠CED=90°, ∠AEC=35°, then the measure of ∠D is ()

Answer: 55°

1. Given AB∥CD, ##according to the parallel axiom 2 of parallel lines, alternate interior angles are equal, we get ∠AEC=∠ECD=35°##.

2. Given ∠CED = 90°, and ##∠ECD## = 35°.

3. ##According to the triangle angle sum theorem##, in △CED, ∠CED + ##∠ECD + ∠EDC## = 180°.

4. ##Substitute the given conditions to get 90° + 35° + ∠EDC = 180°##.

5. ##According to the previous step, we get ∠BED = 180° - 90° - 35° = 55°##.

##7.## Through the above reasoning, the final answer is 55°.

平行线; 同旁内角; 直角三角形

Definition of Alternate Interior Angles

<note>Two parallel lines AB and CD</note> are intersected by <note>a line EC</note>, where <note>angle AEC and angle ECD are located between the two parallel lines and on opposite sides of the intersecting line EC</note>. Therefore, <note>angle AEC and angle ECD are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>angle AEC is equal to angle ECD</note>.

Triangle Angle Sum Theorem

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

<note>In the figure of this problem, in triangle CED, angle CED, angle ECD, and angle EDC are the three interior angles of triangle CED</note>, according to the Triangle Angle Sum Theorem, <note>angle CED + angle ECD + angle EDC = 180°</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.

<note>In the diagram of this problem, two parallel lines AB and CD</note> are intersected by a third line <note>EC</note>, forming the following geometric relationship: <note>alternate interior angles: angle AEC and angle ECD are equal</note>.


Question: As shown in the figure, ⊙O is the circumcircle of △ABC, AB is the diameter, if ∠BOC=70°, then ∠A equals ()

Answer: 35°

1. Given ∠BOC = 70°. (According to the definition of central angle) 2. In a circle, the central angle is equal to twice the inscribed angle that subtends the same arc, i.e., ∠A = 1/2 ∠BOC. (According to the inscribed angle theorem) 3. Based on ∠BOC = 70°, we get ∠A = 1/2 x 70° = 35°. 4. Through the above reasoning, the final answer is 35°.

圆; 圆周角; 圆心角; 圆内接四边形

Definition of Inscribed Angle

In the diagram of this problem, in circle O, <note>the vertex of angle BAC (point A) is on the circumference</note>, <note>the two sides of angle BAC intersect circle O at points B and C respectively</note>. Therefore, angle BAC is an inscribed angle.

Definition of Central Angle

In the figure of this problem, in circle O, <note>points B and C are two points on the circle</note>, <note>the center of the circle is point O</note>. <note>The angle ∠BOC formed by the lines OB and OC is called the central angle</note>.

Inscribed Angle Theorem

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


Question: In the figure above, two line segments meet at a point on line l. If the value of y is equal to the square of the value of x, what is the value of y?

Answer: 100

1. Observe the figure, it is known that x° and y° are the angles formed by two intersecting line segments, where y° equals the square of x°.

2. According to the definition of a straight angle, we can obtain x° + y° + 70° = 180°.

3. Express y in terms of x's square, that is, x + x^2 + 70 = 180.

4. Move 70 to the right side of the equation, we get x + x^2 = 110.

5. Simplify the equation to get x^2 + x - 110 = 0.

6. Solve the equation x^2 + x - 110 = 0 using the quadratic formula, x = [-b ± √(b^2 - 4ac)] / 2a, where a=1, b=1, c=-110.

7. Substitute to obtain the solution: x = [-1 ± √(1 + 440)] / 2, simplify to get x = [-1 ± √(441)] / 2.

8. √(441) = 21, thus x = [-1 + 21] / 2 = 10 or x = [-1 - 21] / 2 = -11, only x = 10 is a positive solution.

9. Further, we get y = x's square, therefore y = 10^2, that is, y = 100.

10. Through the above reasoning, the final answer is y = 100.

线段; 对顶角; 邻补角

Definition of Straight Angle

A straight angle is formed when a ray rotates around its endpoint and the initial side and terminal side lie on the same line but in opposite directions. A straight angle measures 180 degrees.

<note>In the figure of this problem, let the black dot on the line l be point O, one end of the line l be point M, and the other end be point N</note>, <note>the ray l rotates around point O from the end M to form a straight line with the initial side, creating a straight angle</note>. According to the definition of a straight angle, <note>the measure of a straight angle is 180 degrees, that is, angle NOM = 180 degrees</note>.


Question: As shown in the figure, △ABC≌△DEF, points A and D, B and E are corresponding vertices, and it is measured that BC=5cm, BF=7cm, then the length of EC is ()

Answer: 3cm

1. Given △ABC≌△DEF, and point A and point D, point B and point E are corresponding vertices respectively. According to the definition of congruent triangles, corresponding sides are equal, so side BC=side EF=5cm.

2. Given side BF=7cm, side BC=5cm.

3. ##From the figure, we know EC=EF-CF, and because CF=BF-BC=7cm-5cm=2cm, EF=5cm, so EC=EF-CF=5cm-2cm=3cm##.

线段; 普通三角形; 平移

Definition of Congruent Triangles

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

<note>Triangle ABC and triangle DEF are congruent triangles</note>, the corresponding sides and angles of triangle ABC are equal to those of triangle DEF, namely: <note>side AB = side DE</note> <note>side BC = side EF</note> <note>side CA = side FD</note>, and the corresponding angles are also equal: <note>angle A = angle D</note> <note>angle B = angle E</note> <note>angle C = angle F</note>.


Question: Express the ratio of \\cos X as a decimal, accurate to two decimal places.

Answer: 0.28

1. Given conditions: Right triangle VWX, where ∠VWX is a right angle, WV is 72, WX is 21, VX is 75.

2. According to the definition of a right triangle, ∠VWX is a right angle, therefore triangle VWX is a right triangle. Side VW and side WX are the legs, and side VX is the hypotenuse.

3. According to the cosine function: In a right triangle, the cosine of an acute angle is equal to the length of the adjacent side divided by the length of the hypotenuse.

4. Determine ∠VXW as an acute angle, using the cosine function, the cosine of ∠VXW is the length of the adjacent side WX, 21, divided by the length of the hypotenuse VX, 75, i.e., COS(∠VXW) = WX/VX = 21/75.

5. Calculate the value of 21/75 as 0.28.

6. Finally, round the result to two decimal places to get the value of COS(∠VXW) as 0.28.

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 right triangle <note>VWX</note>, side <note>WX is the adjacent side to angle ∠VXW</note>, and side <note>VX is the hypotenuse</note>. Therefore, according to the definition of the cosine function, the cosine value of angle <note>∠VXW is equal to the ratio of the adjacent side WX to the hypotenuse VX</note>, that is, <note>COS(∠VXW) = WX/VX</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 figure of this problem, in triangle VWX, <note>angle ∠VWX is a right angle (90 degrees)</note>, so triangle VWX is a right triangle. <note>Sides WV and WX are the legs</note>, <note>side VX is the hypotenuse</note>.


Question: Find x. Round the side length to the nearest tenth.

Answer: 69.8

1. Observe the given triangle, where one angle is 60° and another angle is 55°.

2. According to the triangle angle sum theorem, the sum of the three interior angles of a triangle is 180°, so the other angle is 180° - 60° - 55° = 65°.

3. Using the sine rule, the formula is: (side length a/ sinA) = (side length b/ sinB) = (side length c / sinC). In this problem, one side is 73, ##and its opposite angle is 65°##. The side length x we are looking for and its opposite angle is 60°.

4. According to the sine rule, x/sin(60°) = 73/sin(65°).

5. By comparison, calculate x = 73 * sin(60°) / sin(65°).

6. Using a calculator, find sin(60°) ≈ 0.8660, sin(65°) ≈ 0.9063.

7. Substitute the sin values into the formula, x ≈ 73 * 0.8660 / 0.9063.

8. Perform the calculation, x ≈ ##69.7539##.

9. Round the value of x to the nearest integer, getting x ≈ 69.8.

10. The final answer is x ≈ 69.8.

普通三角形; 正弦; 余弦

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.

Values of <note>sin(60°)</note> and <note>sin(65°)</note>. <note>sin(60°) = 0.8660</note>, <note>sin(65°) = 0.9063</note>.

Triangle Angle Sum Theorem

In the diagram of this problem, <note>the two interior angles of the triangle are 60° and 55° respectively</note>. According to the <note>Triangle Angle Sum Theorem</note>, <note>the third interior angle = 180° - 60° - 55° = 65°</note>.

Definition of Triangle

<note>A triangle is a geometric figure composed of three non-collinear points and their connecting line segments</note>. The three points are respectively <note>the three vertices of the triangle</note>, and the three line segments are respectively <note>the three sides of the triangle</note>.

Sine Theorem

In any triangle, the ratio of the length of each side to the sine of its opposite angle is constant and equal to the diameter of the circumcircle, i.e., \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2r = D \) (where \( r \) is the radius of the circumcircle, and \( D \) is its diameter).

<note>Let the three vertices of the triangle be A, B, and C, where ∠A = 60°, ∠B = 55°, ∠C = 65°</note>. <note>Side BC = 73, corresponding to angle ∠C = 65°, side BC = x, corresponding to angle ∠A = 60°</note>. According to the Sine Theorem, the ratio of the lengths of the sides to the sine values of their opposite angles are equal, that is: <note>x / sin(60°) = 73 / sin(65°)</note>. Calculated x ≈ 69.8.


Question: As shown in the figure, the perimeter of □ABCD is 16cm. AC and BD intersect at point O. OE⊥AC intersects AD at point E. Then the perimeter of △DCE is ()

Answer: 8cm

1. As shown in the figure, quadrilateral ABCD is a parallelogram with a perimeter of 16 cm.

2. According to the definition of a parallelogram, side AD is equal to side BC, and side AB is equal to side CD. Based on the perimeter of the parallelogram, the perimeter of the parallelogram is equal to twice the sum of its two pairs of adjacent sides. The formula is: P = 2(a + b). Given that sides AD and CD are one pair of adjacent sides, the sum of sides AD and CD is half the perimeter of ABCD, which is 8 cm.

3. Since line segments AC and BD intersect at point O, and OE is perpendicular to AC at point E, ∠AOE = ∠COE = 90°.

4. Because point O is the intersection point of the diagonals of parallelogram ABCD, according to the properties of a parallelogram, the diagonals bisect each other, so AO = CO. Also, since ∠AOE = ∠COE = 90°, triangles △AOE and △COE share side OE. According to the congruence theorem (SAS), the corresponding sides and the included angle of the two triangles are equal, so the two triangles are congruent, thus △AOE ≌ △COE.

5. According to the definition of congruent triangles, the corresponding sides are equal, so AE = EC.

6. To calculate the perimeter of △DCE, we need to find the lengths of line segments DC, DE, and CE, i.e., P = DC + CE + DE.

7. Given AE = EC, AD = AE + DE, so the perimeter of △DCE is P = DC + AE + DE = DC + AD.

8. Given that the sum of sides AD and CD is half the perimeter of ABCD, which is 8 cm, the final perimeter of △DCE is 8 cm.

正方形; 垂线; 线段; 三角形的外角; 普通三角形

Definition of Parallelogram

A quadrilateral is a parallelogram if and only if it has two pairs of opposite sides that are both parallel (∥) and congruent (≅).

<note>Quadrilateral ABCD</note> is a parallelogram, <note>side AD is parallel and equal to side BC</note>, <note>side AB is parallel and equal to side CD</note>.

Perimeter of a Parallelogram

The perimeter of a parallelogram is equal to twice the sum of the lengths of its two adjacent sides. The formula is: \( P = 2(a + b) \), where \( a \) and \( b \) are the lengths of the two adjacent sides of the parallelogram.

In the parallelogram <note>ABCD</note>, sides <note>AD and CD are adjacent sides</note>, and sides <note>AB and BC are also adjacent sides</note>. According to the formula for the perimeter of a parallelogram, the perimeter of the parallelogram is equal to twice the sum of its two pairs of adjacent sides, i.e., <note>Perimeter P = 2(AD + CD)</note>.

Properties of Parallelogram Theorem

In a parallelogram, opposite angles are equal, opposite sides are equal, and the diagonals bisect each other.

<note>In the figure of this problem, in parallelogram ABCD</note>, <note>the opposite angles ∠A and ∠C are equal</note>, <note>the opposite angles ∠B and ∠D are equal</note>; <note>side AB and side CD are equal</note>, <note>side AD and side BC are equal</note>; <note>the diagonals AC and BD bisect each other</note>, that is, <note>the intersection point O divides the diagonal AC into two equal segments AO and OC</note>, <note>the intersection point O divides the diagonal BD into two equal segments BO and OD</note>.

Triangular Congruence Theorem (SAS)

If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

<note>In the figure of this problem, in triangle AOE and triangle COE</note>, <note>side AO is equal to side CO</note>, <note>side OE is equal to side OE</note>, <note>and angle AOE is equal to angle COE</note>, therefore, according to the Triangular Congruence Theorem (SAS), these two triangles are congruent.

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 \( P = a + b + c \).

The three sides of <note>triangle DCE</note> are <note>CD, CE, DE</note> respectively. According to the <note>Formula for the Perimeter of a Triangle</note>, the perimeter of the triangle is equal to the sum of the lengths of its three sides, that is, the perimeter <note>P=CD+CE+DE</note>.

Definition of Congruent Triangles

<note>Triangle AOE and triangle COE are congruent triangles</note>, the corresponding sides and corresponding angles of triangle AOE are equal to those of triangle COE, namely:
<note>side AO = side OC</note>
<note>side AE = side CE</note>
<note>side OE = side OE</note>
At the same time, the corresponding angles are also equal:
<note>angle AOE = angle COE</note>
<note>angle AEO = angle CEO</note>
<note>angle EAO = angle ECO</note>.


Question: Find m \angle N C L.

Answer: 120

2. In this problem's figure, according to the definition of adjacent supplementary angles, ∠NCL and ∠MCN share a common side NC, and their other sides CL and CM are extensions in opposite directions, thus they are adjacent supplementary angles.

3. According to the definition of a straight angle, ∠NCL + ∠MCN = 180°, therefore it can be concluded that ∠NCL = 180° - ∠MCL.

4. Substitute the value ∠MCN = 60°, obtaining ∠NCL = 180° - 60°.

5. Calculate 180° - 60°, obtaining ∠NCL = 120°.

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

圆心角; 圆周角; 普通三角形

Definition of Linear Pair of Angles

Two angles are in a linear pair if they share a common side, and their other sides are extensions of one another in opposite directions.

<note>Angle NCL</note> and <note>Angle MCN</note> have a common side <note>NC</note>, and their other sides <note>CL</note> and <note>CM</note> are extensions in opposite directions, so angle NCL and <note>MCN</note> are adjacent supplementary angles, therefore ∠NCL + <note>∠MCN</note> = 180°.

Definition of Straight Angle

<note>Ray MC rotates around endpoint C until it forms a straight line with the initial side</note>, forming a <note>straight angle MCL</note>. According to the definition of a straight angle, the measure of a straight angle is 180 degrees, i.e., <note>angle MCL = 180 degrees</note>.


Question: What is the area of the sector shown above?

Answer: 49*\pi/3

1. Assume the sector is formed by the center O, radius OA and OB, and arc AB. The given condition is that the radius OA is 7, and the central angle AOB is 120 degrees.

2. According to the sector area formula, sector area = π × r² × (θ / 360), where r is the radius and θ is the central angle in degrees. Substitute the given conditions into the sector area formula to get: sector area = π × 7² × (120 / 360).

3. First, calculate the square of 7, which is 7² = 49.

4. Calculate the angle ratio, 120 / 360 = 1 / 3.

5. Therefore, sector area = π × 49 × 1 / 3 = 49π / 3.

6. Through the above reasoning, the final answer is 49π / 3.

Definition of Sector

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

<note>A sector is formed by the center O</note>, <note>radii OA and OB</note>, and <note>arc AB</note>. <note>The central angle ∠AOB = 120°</note>, <note>radius OA = OB</note> = 7.

Definition of Central Angle

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

Formula for the Area of a Sector

The area \( X \) of a sector can be calculated using the formula \( X = \frac{\theta}{360} \times \pi \times r^2 \), where \( \theta \) is the measure of the central angle in degrees, and \( r \) is the radius length.

<note>In the figure of this problem, the radius of the sector r = 7, and the central angle θ = 120°</note>, according to the formula for the area of a sector, the area A of the sector can be calculated using the formula <note>A = (θ/360) * π * r2</note>, where <note>θ is the measure of the central angle</note>, <note>r is the length of the radius</note>. Therefore, <note>the area of the sector A = (120/360) * π * 72</note>.

Definition of Radius

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

In the figure of this problem, in circle O, <note>point O is the center of the circle</note>, <note>point A is any point on the circle</note>, <note>line segment OA is the line segment from the center to any point on the circle</note>, therefore <note>line segment OA is the radius of the circle</note>.

Definition of Arc

An arc is a curve on a circle that is defined by two distinct points on the circle.

<note>There are two points A and B on circle O</note>, arc AB is a segment of the curve connecting these two points. According to the definition of arc, <note>arc AB is a segment of the curve between two points A and B on the circle</note>.


Question: Find x.

Answer: 7

1. From the figure, it can be seen that triangle RQS has an inscribed circle with center W. According to the inscribed circle, the distances from center W to the three sides of triangle RQS are equal. The points where sides RQ and QS are tangent to circle W are points U and T, respectively, so UW ⊥ QR and TW ⊥ QS, i.e., ∠WUQ = ∠WTQ = 90°. Connecting UW, TW, and QW forms triangles QWU and QWT. According to the definition of right triangles, since ∠WUQ = ∠WTQ = 90°, triangles QWU and QWT are both right triangles.

2. According to the criteria for congruence of right triangles (hypotenuse and a leg), in right triangles QWU and QWT, UW = TW. Sides UW and TW are the legs of the two right triangles, and QW is the common hypotenuse, so right triangle QWU is congruent to right triangle QWT.

3. According to the definition of congruent triangles, QU = QT. From the figure, it is noted that QT = 2x in., QU = 14 in., so 2x = 14.

5. Through the above reasoning, we can definitively conclude that the final answer is 7.

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 \( \triangle RQS \)</note> is a geometric figure composed of three non-collinear points <note>\( R \)、\( Q \)、\( S \)</note> and their connecting line segments <note>\( RQ \)、\( QS \)、\( RS \)</note>. <note>Points \( R \)、\( Q \)、\( S \)</note> are the three vertices of the triangle, and <note>line segments \( RQ \)、\( QS \)、\( RS \)</note> are the three sides of the triangle.

Definition of Circle

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

<note>In circle W, point W is the center</note>. In the figure, <note>all points that are at a distance equal to the radius from point W are on circle C</note>.

Incircle

An incircle of a polygon is a circle that is tangent to each side of the polygon. The center of this circle is called the incenter, and the distance from the incenter to each side of the polygon is equal.

In the figure of this problem, <note>circle \( W \) is tangent to the sides \( RQ \), \( QS \), and \( RS \) of triangle \( \triangle RQS \)</note>, so <note>circle \( W \) is the incircle of \( \triangle RQS \)</note>.

Definition of Right Triangle

<note>Angle WUQ is a right angle (90 degrees)</note>, therefore triangle QUW is a right triangle. Side <note>WU</note> and side <note>QU</note> are the legs, and side <note>QW</note> is the hypotenuse. <note>Angle WTQ is a right angle (90 degrees)</note>, therefore triangle QTW is a right triangle. Side <note>WT</note> and side <note>QT</note> are the legs, and side <note>QW</note> is the hypotenuse.

Two right triangles are congruent if their hypotenuses and one corresponding leg are equal.

<note>In right triangle QUW and right triangle QTW</note>, suppose <note>angle WUQ and angle WTQ are both right angles (90 degrees)</note>. If <note>side QW is equal to side QW (hypotenuse)</note>, and <note>side WU is equal to side TW (one leg)</note>, then according to <note>Right Triangle Congruence (Hypotenuse, One Leg)</note>, these <note>two right triangles QUW and QTW are congruent</note>.

Definition of Congruent Triangles

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

<note>Triangle QWT and triangle QUT are congruent triangles</note>, the corresponding sides and angles of triangle QWT are equal to those of triangle QUT, namely:
side <note>QU = side QT</note>
side <note>QW = side QW</note>
side <note>WU = side WT</note>, meanwhile, the corresponding angles are also equal:
<note>angle WUQ = angle WTQ</note>
<note>angle WQU = angle WQT</note>
<note>angle QWU = angle QWT</note>.


Question: Find the value of x.

Answer: 23

1. Let the points on the circle be A, B, C, D, and the center of the circle be point O.

2. Observing the figure, it is known that the red mark indicates a right angle symbol, which means ∠AOB is a 90-degree right angle.

3. It is known that the central angle ∠BOC = 67°, ∠AOB = 90° in the figure.

4. Since points D, O, C are on the same straight line, according to the definition of a straight angle, ∠BOC + ∠AOB + x = 180°, that is, x = 180° - ∠BOC - ∠AOB = 180° - 67° - 90° = 23°.

6. Therefore, through the above analysis, the final answer is x = 23°.

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 circle O, points B and C are two points on the circle, and the center of the circle is point O. The angle ∠BOC formed by the lines OB and OC is called the central angle.</note>

<note>Ray OD rotates around endpoint O until it forms a straight line with the initial side, forming a straight angle DOC</note>. According to the definition of a straight angle, the measure of a straight angle is 180 degrees, i.e., <note>angle DOC = 180 degrees</note>.


Question: Find the value of x.

Answer: 4 \sqrt { 15 }

1. Given that the figure is a right triangle, where one of the legs is 4 and the hypotenuse is 16. Let the vertex where the hypotenuse 16 and the leg X meet be A, let the vertex of the 90° angle be B, and let the vertex where the hypotenuse 16 and the leg 4 meet be C.

2. According to the properties of a right triangle, specifically the Pythagorean theorem, the theorem states: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. The mathematical expression is AC² = AB² + BC².

3. Substitute the given conditions into the Pythagorean theorem: 16² = x² + 4².

4. Calculate: 256 = x² + 16.

5. Solve the equation to find the value of x: 256 - 16 = x², which is 240 = x².

6. Finally, the result for x is: x = √240 = 4√15.

7. Since x represents a side length, and side lengths cannot be negative, x = 4√15.

Definition of Right Triangle

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

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

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 context of a <note>right triangle</note>, <note>one leg length is 4</note>, <note>the hypotenuse length is 16</note>, and <note>the other leg length is x</note>. Therefore, according to the <note>Pythagorean Theorem</note>, ##16² = 4² + x²##, which means 256 = 16 + x², solving for <note>x² = 240</note>, thus <note>x = √240, which is 4√15</note>.


Question: Find the perimeter of the figure.

Answer: 32

1. The given condition is a right trapezoid, ##where the lower base is 4, the height is 8, and the upper base is 10. Let this right trapezoid be right trapezoid ABCD##.

2. ##Draw a line segment DE from point D upwards perpendicular to the upper base AB. Then AE, DE, and AD form triangle AED. Since DE⊥AB, ∠DEA=∠DEB=90°, so triangle AED is a right triangle and DE≌BC##.

3. ##Also, since ∠DEB=90°, quadrilateral EBCD is a rectangle. According to the definition of a rectangle, opposite sides are equal, so EB=CD=4. Also, since AE+EB=AB=10, AE=AB-EB=10-4=6##.

4. ##In right triangle AED, according to the Pythagorean theorem, AD^2=AE^2+DE^2. Given AE=6, DE=BC=8, so AD^2=36+64=100, solving: AD=10##.

5. ##According to the formula for the perimeter of a polygon, the perimeter of a polygon is equal to the sum of the lengths of all its sides. So the perimeter of trapezoid ABCD P=AB+BC+CD+AD=10+8+4+10=32##.

####

##6##. Through the above reasoning, the final answer is 32.

普通多边形; 直角三角形; 线段

Right Angle Trapezoid

A trapezoid is classified as a right angle trapezoid if and only if its two bases are parallel (∥), and one of its non-parallel sides (leg) is perpendicular (⊥) to the bases.

<note>In the right angle trapezoid ABCD</note>, <note>the base AB and the base CD are parallel</note>, <note>the leg BC is perpendicular to the base AB and the base CD</note>, <note>angle ABC and angle BCD are right angles</note>. Therefore, <note>the trapezoid ABCD is a right angle trapezoid</note>.

Definition of Triangle

<note>Triangle AED</note> is a geometric figure formed by <note>three non-collinear points A, E, D</note> and their connecting line segments <note>AE, AD, ED</note>. <note>Points A, E, D are the three vertices of the triangle</note>, and <note>line segments AE, AD, ED are the three sides of the triangle</note>.

Definition of Right Triangle

<note>In the diagram of this problem, in triangle AED, angle AED is a right angle (90 degrees), so triangle AED is a right triangle</note>.<note>Side AE and side ED are the legs</note>,<note>side AD is the hypotenuse</note>.

Pythagorean Theorem

In the figure of this problem, <note>in the right triangle AED</note>, <note>angle AED is a right angle (90 degrees)</note>, <note>sides AE and ED are the legs</note>, <note>side AD is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>AD² = AE² + ED²</note>.

Definition of Rectangle

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

<note>Quadrilateral EBCD is a rectangle</note>, its interior angles <note>∠EBC, ∠BCD, ∠CDE, ∠DEB are all right angles (90 degrees)</note>, and <note>side ED is parallel and equal in length to side BC</note>, <note>side EB is parallel and equal in length to side CD</note>.


Question: As shown in the figure, a cylinder with a base circumference of 24m and a height of 5m, the shortest path an ant travels along the surface from point A to point B is ()

Answer: 13m

1. ##Unfold the lateral surface of the cylinder into a quadrilateral. Since the top and bottom surfaces of the cylinder are the same, one pair of opposite sides of the quadrilateral is the circumference of the base circle = 24m, and the other pair of opposite sides is the height of the cylinder = 5m. The two pairs of opposite sides are parallel to each other. According to the definition of a rectangle, the unfolded lateral surface of the cylinder is a rectangle. The width of the rectangle is 5m, and the length of the rectangle is the circumference of the base circle, 24m.##

2. ##According to the diagram, points A and B are on opposite sides of the cylinder, located at the top and bottom vertices respectively. That is, point A is at one vertex of the unfolded rectangle, and point B is at the midpoint of the bottom edge of the rectangle.##

3. ##Let the parallel distance from point B to point A be BO, then the height from point A to point O, AO, is 5m. AO is perpendicular to BO, i.e., ∠AOB = 90°. According to the definition of a right triangle, triangle AOB is a right triangle; BO = circumference of the base circle / 2 = 24 / 2 = 12m.##

4. ##Using the Pythagorean theorem, we can derive that in triangle AOB, AB^2 = BO^2 + AO^2.##

5. ##Substitute the known values: AB = √(BO^2 + AO^2) = √(12^2 + 5^2) = √(144 + 25) = √169 = 13m.##

6. Therefore, the shortest path that the ant travels along the surface from point A to point B is 13m.

圆柱; 平移; 圆; 直角三角形

Cylinder

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

In this problem diagram, <note>a cylinder consists of two parallel and identical circular bases and a lateral surface</note>. <note>The circumference of the cylinder's base is 24 meters</note>, and the height is 5 meters. <note>The two circular bases of the cylinder are the circular planes where point A and point B are located, respectively</note>. <note>The lateral surface of the cylinder, when unfolded, forms a rectangle with a height of 5 meters and a width of 24 meters</note>.

Definition of Rectangle

<note>After the cylinder is unfolded, the resulting quadrilateral is a rectangle</note>, whose <note>internal angles are all right angles (90 degrees)</note>, and <note>opposite sides are parallel and equal in length</note>. The rectangle's <note>height is 5 meters</note>, and <note>the length of the base is the circumference of the cylinder's base, 24 meters</note>. <note>Point A and point B are located at the vertex and the midpoint of the side length of the rectangle, respectively</note>.

Definition of Right Triangle

Definition of Right Triangle:
'''A triangle with one <note>interior angle of 90 degrees</note> is called a <note>right triangle.</note>'''

<note>Key</note> description:
'##In the problem diagram, by unfolding the <note>cylinder's side to form a rectangle</note>, the shortest route from <note>point A to point B</note> is described as a diagonal line across the <note>rectangle</note>, forming a <note>right triangle</note>, where <note>AO is one leg</note>, <note>OB is the other leg</note>, and <note>AB is the hypotenuse.</note>##'

Pythagorean Theorem

In the figure of this problem, the shortest route from point A to point B is required. By unfolding the lateral surface of the cylinder to form a rectangle and using the <note>formed right triangle</note>, the <note>Pythagorean Theorem</note> can be applied. Suppose the height of the rectangle is 5 meters, <note>half of the rectangle's base is 12 meters</note>, according to the <note>Pythagorean Theorem, the length of the hypotenuse</note> is: √(12^2 + 5^2) = √(144 + 25) = √169 = 13m.


Question: Find m \angle A B C.

Answer: 51

1. Given the condition: The angle formed by the extension of line BC in the direction of point C and segment AC is 148°, and the interior angles of the triangle are ∠BAC = (2x-15)°, ∠ABC = (x-5)°, we need to find ∠ABC. According to the exterior angle theorem of triangles, since the 148° angle is the exterior angle of ∠ACB in triangle ACB, we have ∠BAC + ∠ABC = 148°.

2. From the figure: ∠BAC = (2x-15)°, ∠ABC = (x-5)°, substituting into the relationship equation we get: 2x-15 + x-5 = 148.

3. Solving the equation: 3x-20 = 148, combining the numbers we get: 3x = 168, finally we get: x = 56.

4. Therefore, when x = 56, the value of ∠ABC is 51°.

5. Finally, according to the problem, the degree of ∠ABC is 51°.

普通三角形; 三角形的外角

Definition of Triangle

In the figure of this problem, <note>triangle ABC is a geometric figure composed of three non-collinear points A, B, C and their connecting line segments AC, AB, BC</note>. <note>Points A, B, C are the three vertices of the triangle</note>, <note>line segments AC, AB, BC are the three sides of the triangle</note>.

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

The exterior angle of the interior angle <note>∠ACB</note> of <note>triangle ABC</note> is <note>148°</note>. The <note>148°</note> angle formed by extending <note>side BC</note> is called the exterior angle of the interior angle <note>∠ACB</note>.

Exterior Angle Theorem of Triangle

An exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

<note>The 148° angle is an exterior angle of a triangle</note>, <note>angle BAC and angle ABC are the two non-adjacent interior angles to the 148° exterior angle</note>, according to the Exterior Angle Theorem of Triangle, <note>the 148° angle is equal to the sum of the two non-adjacent interior angles BAC and ABC</note>, that is, <note>148° angle = angle BAC and angle ABC</note>.


Question: Find x.

Answer: 8 \sqrt { 5 }

1. The given condition is a right triangle, where one leg is 38, the other leg is x, and the hypotenuse is 42.

2. According to the Pythagorean theorem, in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse, i.e., a^2 + b^2 = c^2, where b and a are the legs, and c is the hypotenuse.

3. Substitute the given values into the Pythagorean theorem, obtaining the equation x^2 + 38^2 = 42^2.

4. Calculate 38^2 = 1444.

5. Calculate 42^2 = 1764.

6. Substitute the calculation results, obtaining the equation x^2 + 1444 = 1764.

7. Solve the equation x^2 + 1444 = 1764, obtaining x^2 = 1764 - 1444.

8. Calculate 1764 - 1444 = 320.

9. Obtain the equation x^2 = 320, taking the positive value of x, obtaining x = √320.

10. Further calculate, √320 = ##√(64×5) = 8 × √5 = 8√5##.

11. Since the radical form is retained, my final answer is x = ##8√5##.

12. Through the above reasoning, the final answer is ##8√5##.

Definition of Right Triangle

<note>The right angle is located in the upper right corner</note>, <note>the legs are 38 and x</note>, <note>the hypotenuse is 42</note>.

Pythagorean Theorem

In the figure of this problem, in the <note>right-angled triangle</note>, one <note>leg is 38</note>, the other <note>leg is x</note>, and the <note>hypotenuse is 42</note>. According to the <note>Pythagorean Theorem</note>, the square of the hypotenuse is equal to the sum of the squares of the two legs, that is, <note>38^2 + x^2 = 42^2</note>.


Question: Find y.

Answer: 10

1. ##Let the three vertices of the right triangle be A, B, and C, where ∠A=60°, ∠C=90°, AB=20. According to the definition of a right triangle, this triangle is a right triangle. According to the triangle sum theorem, ∠B=30°. Thus, it can be understood that this is a 30°-60°-90° triangle.##

2. ##According to the properties of a 30°-60°-90° triangle, the ratio of the three sides is 1:√3:2.##

3. ##To find the length of y, use the ratio of y to the hypotenuse, which is 1:2. Therefore, y=hypotenuse/2.##

4. ##Substitute the given hypotenuse of 20 into the equation, and calculate y = 20 / 2 = 10.##

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

直角三角形; 正弦; 余弦

Definition of Right Triangle

In the figure of this problem, <note>one of the interior angles of the triangle is 90 degrees</note>, so the triangle is a right triangle. <note>y and x are the legs, and the hypotenuse is 20</note>.

Triangle Angle Sum Theorem

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

Original text: <note>∠A=60°</note>, <note>∠C=90°</note>, therefore <note>∠B=30°</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>∠B=30°</note>, <note>∠A=60°</note>, <note>∠C=90°</note>. Side <note>AB is the hypotenuse</note>, side <note>AC is the side opposite the 30-degree angle</note>, side <note>BC is the side opposite the 60-degree angle</note>. y is equal to half of the hypotenuse, x is equal to y multiplied by √3. That is: <note>AC = 1/2 * AB</note>, <note>BC = AC * √3</note>.


Question: Quadrilateral W X Y Z is a rectangle. If m \angle 1 = 30, find the measure of \angle 6.

Answer: 60

1. Given WXYZ is a rectangle, ##according to the definition of a rectangle: WX∥YZ, WY∥XZ, and according to the definition of alternate interior angles and the parallel postulate 2, ∠1=∠5=30° (alternate interior angles are equal)##.

##2. From the right angle mark, we know that ∠WYZ = 90 degrees.##

##3. From the figure, we get ∠6+∠5=∠WYZ, now we know ∠5=30°, ∠WYZ=90°, then ∠6=∠WYZ-∠5=90°-30°=60°.##

####

##4##. Through the above reasoning, we finally get the answer as 60 degrees.

矩形; 垂线; 邻补角; 内错角

Definition of Rectangle

<note>Quadrilateral WXYZ is a rectangle</note>, with interior angles <note>∠WYZ, ∠YWX, ∠WXZ, ∠XZY</note> all being right angles (90 degrees), and <note>side WX is parallel and equal in length to side YZ, side WY is parallel and equal in length to side XZ</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 WX and YZ are intersected by a transversal XY,</note><note>where ∠1 and ∠5 are located between the two parallel lines and on opposite sides of the transversal XY,</note><note>therefore ∠1 and ∠5 are alternate interior angles.</note><note>Alternate interior angles are equal, that is, ∠1 is equal to ∠5.</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, <note>two parallel lines WY and XZ</note> are <note>intersected by a line XY</note>, forming the following geometric relationships: <note>alternate interior angles: ∠1 and ∠5 are equal</note>. These relationships indicate that when two parallel lines are intersected by a third line, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary.


Question: As shown in the figure, line AB∥CD, ∠C=44°, ∠E is a right angle, then ∠1 equals ()

Answer: 134°

1. Draw FE through point E parallel to AB. Given AB∥CD, then FE∥CD.

2. According to the definition of alternate interior angles, ∠FEC and ∠C are alternate interior angles, ∠BAE and ∠AEF are alternate interior angles. According to the parallel axiom 2 of parallel lines, ∠FEC = ∠C, ∠BAE = ∠AEF.

3. Given ∠C = 44°, we can obtain ∠FEC = 44°.

4. From the diagram and given conditions, ∠AEC = ∠AEF + ∠FEC = 90°, then substituting the given conditions, we get ∠AEF = ∠AEC - ∠FEC = 90° - 44° = 46°.

5. According to step 2, we get ∠AEF = ∠BAE = 46°.

6. According to the definition of a straight angle, the angle formed at point A is a straight angle, i.e., ∠1 + ∠BAE = 180°.

8. Through the above reasoning, the final answer is 134°.

平行线; 同位角; 内错角; 直角三角形

Definition of Alternate Interior Angles

<note>Two parallel lines CD and FE</note> are intersected by <note>a line CE</note>, where <note>∠FEC and ∠C</note> are located between the two parallel lines and on opposite sides of the intersecting line <note>CE</note>, thus <note>∠FEC and ∠C are alternate interior angles</note>; <note>Two parallel lines AB and FE</note> are intersected by <note>a line AE</note>, where <note>∠BAE and ∠AEF</note> are located between the two parallel lines and on opposite sides of the intersecting line <note>AE</note>, thus <note>∠BAE and ∠AEF are alternate interior angles</note>.

Parallel Postulate 2 of Parallel Lines

Original: <note>EF平行于AB</note>, line <note>AE</note> intersects these two parallel lines, forming the following geometric relationship: <note>Alternate interior angles: ∠BAE and ∠AEF are equal</note>. Similarly, <note>EF平行于CD</note>, line <note>CE</note> intersects these two parallel lines, forming the following geometric relationship: <note>Alternate interior angles: ∠FEC and ∠C are equal</note>.

Definition of Straight Angle

A straight angle is formed when a ray rotates around its endpoint and the initial side and terminal side lie on the same line but in opposite directions. A straight angle measures 180 degrees.

<note>The ray rotates around endpoint A until it forms a straight line with the initial side</note>, forming a straight angle. According to the definition of a straight angle, <note>the measure of a straight angle is 180 degrees</note>, that is, <note>the straight angle formed at point A is 180 degrees</note>.

Transitivity of Parallel Lines

If two lines are each parallel to a third line, then those two lines are parallel to each other.

<note>Line EF and line CD are respectively parallel to line AB</note>. According to the transitivity of parallel lines, if <note>line EF is parallel to line AB</note>, and <note>line CD is also parallel to line AB</note>, then <note>line EF and line CD are parallel to each other</note>. Therefore, <note>line EF is parallel to line CD</note>.


Question: AB is the diameter, AC = 8 inches, BC = 15 inches. Find the circumference of the circle.

Answer: 17 \pi

1. First, it is known that AB is the diameter of the circle. Let the center of the circle be O, so O is the midpoint of AB, and OA = OB is the radius of the circle.

2. According to the definition of the inscribed angle and the corollary of the inscribed angle theorem, the inscribed angle subtended by the diameter is a right angle because AB is the diameter. Here, it can be deduced that ∠ACB is a right angle.

3. According to the definition of a right triangle, since ∠ACB = 90°, △ABC is a right triangle. In the right triangle ACB, we know AC = 8 inches and BC = 15 inches. Therefore, using the Pythagorean theorem, we can find the length of AB. The Pythagorean theorem states: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs, that is, AB^2 = AC^2 + BC^2.

4. Calculate AB: AB^2 = 8^2 + 15^2 = 64 + 225 = 289. Therefore, AB = √289 = 17 inches.

5. AB is the diameter, so the diameter of the circle is 17 inches.

6. The formula for the circumference of a circle is C = 2πr, where r is the radius. Since the diameter of the circle is twice the radius, the formula for the circumference can also be C = πd = π × 17 = 17π inches, where d is the diameter.

7. Finally, the circumference of the circle is calculated to be 17π inches.

圆; 直角三角形; 线段

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>In circle O, point O is the center of the circle</note>, <note>line segment AO is the radius r</note>. According to the circumference formula of the circle, the circumference C of the circle is equal to 2π multiplied by the radius r, that is, <note>C=2πr</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 ACB (point C) is on the circumference</note>, and <note>the two sides of angle ACB intersect circle O at points A and B, respectively</note>. Therefore, <note>angle ACB is an inscribed angle</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 circle O, the diameter AB subtends a right angle (90 degrees) at the circumference angle ACB.

Definition of Right Triangle

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

Pythagorean Theorem

<note>In the right triangle ACB</note>, <note>∠ACB is a right angle (90 degrees)</note>, the sides <note>AC and BC are the legs</note>, and the side <note>AB is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>AB^2 = AC^2 + BC^2 = 8^2 + 15^2 = 64 + 225 = 289</note>, therefore <note>AB = √289 = 17 inches</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>In the figure of this problem, AB is the diameter</note>, connecting <note>the center O and the points A and B on the circumference</note>, with a length of <note>twice the radius, that is, AB = 2AO = 2BO</note>.


Question: Shaoxing is famous for its bridges. As shown in the figure, the distance from the top of the arch bridge to the water surface CD is 8m, and the radius of the arch OC is 5m. Then the width of the water surface AB is ()

Answer: 8m

1. Connect point O to point A.

2. Since CD is perpendicular to AB, according to the given condition AD equals BD and the diameter theorem, we can get AD equals BD equals half of AB.

3. In the right triangle ADO, OA equals OC which is 5 meters, OD equals CD minus OC, that is, OD equals 8 meters minus 5 meters, which equals 3 meters.

4. Since angle ADO is 90 degrees, according to the Pythagorean theorem, in the right triangle ADO, AD equals √(OA^2 - OD^2) = √(5^2 - 3^2) = 4 meters.

5. Therefore, AB equals 2 times AD, that is, AB equals 2 × 4 meters = 8 meters.

圆; 弦; 线段; 垂线; 直角三角形

Definition of Radius

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

In the figure of this problem, <note>in circle O, point O is the center of the circle</note>, <note>points A and C are any points on the circle</note>, <note>segments OA and OC are segments from the center to any point on the circle</note>, therefore <note>segments OA and OC are the radius of the circle</note>, and <note>OA equals OC equals 5 meters</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 figure of this problem, in triangle ADO, <note>angle ADO is a right angle (90 degrees)</note>, therefore <note>triangle ADO is a right triangle</note>. <note>Side OD and side AD are the legs</note>, <note>side OA is the hypotenuse</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.

<note>In the figure of this problem, in the right triangle ADO, ∠ADO is a right angle (90 degrees)</note>, <note>sides OD and AD are the legs</note>, <note>side OA is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>##OA^2= OD^2+ AD^2##</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 <note>circle O</note>, the <note>line containing the diameter CD is perpendicular to chord AB</note>, then according to the <note>Perpendicular Diameter Theorem</note>, <note>the diameter CD bisects chord AB, AD=BD</note>, and <note>bisects the two arcs subtended by chord AB</note>.


Question: As shown in the figure, AB∥CD, CP intersects AB at O, AO=PO, if ∠C=50°, then the degree of ∠A is ()

Answer: 25°

1. Given AB∥CD, CP intersects AB at O, according to the ##parallel lines axiom 2, corresponding angles are equal##. Therefore, ∠POB=∠C.

2. According to the given condition ∠C=50°, we can get ∠POB=50°.

3. Since AO=PO, according to the definition of an isosceles triangle, triangle AOP is an isosceles triangle. Therefore, ∠OAP=∠A.

4. In triangle AOP, ##according to the exterior angle theorem of triangles, ∠A + ∠P = ∠POB##.

5. Because AO=PO, ∠A=∠P, so it can be written as ##2∠A = ∠POB##.

6. Substitute the given ∠POB = 50°, we get ##2∠A =50°##.

7. Solve this equation, ##get ∠A = 25°##.

8. ####Therefore, the correct answer is 25°.

平行线; 同位角; 内错角; 等腰三角形

Definition of Isosceles Triangle

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

<note>In the figure of this problem, in triangle AOP, side AO and side PO are equal</note>, therefore triangle AOP is an isosceles triangle.

Parallel Postulate 2 of Parallel Lines

In the figure of this problem, <note>two parallel lines AB and CD are intersected by a third line CP</note>, forming the following geometric relationships: 1. Corresponding angles: <note>∠POB and ∠C are equal</note>. 2. Alternate interior angles: <note>∠AOC and ∠C are equal</note>. 3. Consecutive interior angles: <note>∠BOC and ∠COD are supplementary</note>, that is, <note>∠BOC + ∠COD = 180 degrees</note>. These relationships illustrate that when <note>two parallel lines are intersected by a third line</note>, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.

Triangle Angle Sum Theorem

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

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.

<note>In the isosceles triangle AOP, side AO and side PO are equal</note>. Therefore, according to the properties of isosceles triangles, <note>the angles opposite the equal sides are equal</note>, i.e., <note>angle A = angle P</note>.

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.

In the figure of this problem, <note>two parallel lines AB and CD</note> are intersected by <note>a line PC</note>, where <note>angle POB and angle C are located on the same side of the intersecting line PC, on the same side of the intersected lines AB and CD</note>, therefore <note>angle C and angle POB are corresponding angles</note>. <note>Corresponding angles are equal, that is, angle C is equal to angle POB</note>.

Exterior Angle Theorem of Triangle

An exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

In the figure of this problem, in triangle AOP, <note>angle POB is an exterior angle of the triangle</note>, <note>angle A and angle P are the two interior angles that are not adjacent to the exterior angle POB</note>. According to the Exterior Angle Theorem of the triangle, <note>the exterior angle POB is equal to the sum of the two non-adjacent interior angles A and P</note>, that is, <note>angle POB = angle A + angle P</note>.

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

The original Chinese enclosed in <note>多边形OAP的一个内角为∠AOP</note>, extending the adjacent sides of this interior angle <note>AP和PO形成的角∠POB</note> is called the exterior angle of the interior angle ∠AOP.


Question: If DB = 24, AE = 3, and EC = 18, find AD.

Answer: 4

1. Given DE parallel to BC, according to the parallel axiom 2 of parallel lines, ∠ADE = ∠B.

2. Since ∠ADE = ∠B and ∠A is shared, according to the AA criterion of similar triangles, △ADE and △ABC are similar.

3. According to the definition of similar triangles, AD/AB = AE/AC, i.e., AD/(AD+24) = 3/(3+18).

4. Solving this equation, AD = 4.

5. Through the above reasoning, it is confirmed that the calculation is correct, and AD = 4.

Similarity Theorem for Triangles (AA)

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

In the figure of this problem, triangle <note>ADE</note> and triangle <note>ABC</note> have angle <note>ADE = angle B</note>, and <note>angle A is shared</note>, so triangle <note>ADE and triangle ABC are similar</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 diagram of this problem, <note>two parallel lines DE and BC</note> are intersected by <note>a third line AB</note>, forming the following geometric relationship: corresponding angles: <note>angle ADE = angle B</note>. These relationships illustrate that when two parallel lines are intersected by a third line, the corresponding angles are equal.

Definition of Corresponding Angles

In the figure of this problem, <note>two parallel lines DE and BC are intersected by a third line AB</note>, where <note>angle ADE and angle B are located on the same side of the intersecting line AB</note>, <note>on the same side of the intersected lines DE and BC</note>, therefore <note>angle ADE and angle B are corresponding angles</note>. Corresponding angles are equal, i.e., <note>angle ADE = angle B</note>.

Definition of Similar Triangles

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

<note>Triangle ADE and triangle ABC are similar</note>. According to the definition of similar triangles: <note>AD/AB = AE/AC</note>.


Question: Express the ratio of \tan L as a decimal, accurate to two decimal places.

Answer: 0.42

1. In triangle LJK, ∠JKL is a right angle. According to the definition of a right triangle, △LJK is a right triangle.

2. After confirming the right triangle, the required value of tan ∠L can be determined. According to the definition of the tangent function, tan ∠L = JK / LK = 5 / 12.

3. Calculating the ratio 5/12, we get tan ∠L ≈ 0.41666...

4. Keeping the result to the second decimal place and rounding, tan ∠L = 0.42.

5. Through the above reasoning, the final answer is that the value of tan ∠L is 0.42.

Definition of Right Triangle

In the figure of this problem, in triangle LJK, <note>angle ∠JKL is a right angle (90 degrees)</note>, therefore <note>triangle LJK is a right triangle</note>. <note>Side JK and side LK are the legs, and side LJ is the hypotenuse</note>.

Definition of Tangent Function

In a right-angled triangle, the tangent value of an acute angle is equal to the length of the opposite side divided by the length of the adjacent side.

In the right triangle <note>LJK</note>, <note>angle ∠L is an acute angle</note>, <note>side JK is the opposite leg of angle ∠L</note>, <note>side LK is the adjacent leg of angle ∠L</note>, so <note>the tangent value of angle ∠L is equal to the length of side JK divided by the length of side LK</note>, i.e., tan(<note>∠L</note>) = <note>JK / LK</note> = 5 / 12.


Question: Find y such that the quadrilateral is a parallelogram.

Answer: 14

1. According to the problem statement, assume the quadrilateral in the figure is a parallelogram, with the four vertices being A, B, C, and D. According to the properties theorem of parallelograms, opposite angles are equal, so angle CAB = 8x - 8° and angle CDB = 6x + 14°, thus 8x - 8 = 6x + 14.

2. Simplifying, we get 8x - 6x = 14 + 8, and after rearranging, we get 2x = 22, thus x = 11.

3. Also according to the properties theorem of parallelograms, angle ABD = 7y + 2° and angle DCA = 6y + 16°, thus 7y + 2 = 6y + 16.

4. Simplifying, we get 7y - 6y = 16 - 2, and after rearranging, we get y = 14.

5. After the above reasoning, the final answer is y = 14.

平行四边形; 对顶角; 邻补角; 同旁内角

Definition of Parallelogram

A quadrilateral is a parallelogram if and only if it has two pairs of opposite sides that are both parallel (∥) and congruent (≅).

In the figure of this problem, the quadrilateral <note>ABCD is a parallelogram</note>, <note>sides AB and CD are parallel and equal</note>, <note>sides AC and BD are parallel and equal</note>.

Properties of Parallelogram Theorem

In a parallelogram, opposite angles are equal, opposite sides are equal, and the diagonals bisect each other.

The original text: <note>Angle ABD and angle DCA</note> are equal, <note>Angle CAB and angle CDB</note> are equal; opposite sides are equal, diagonals bisect each other.

Definition of Quadrilateral

A quadrilateral is a closed geometric figure composed of four line segments, which are referred to as the sides of the quadrilateral. It has four vertices and four interior angles.

<note>Quadrilateral ABCD is composed of four line segments AB, CD, AC, and BD</note>, these line segments are called the sides of the quadrilateral. <note>Quadrilateral ABCD has four vertices, which are point A, point B, point C, and point D</note>, and <note>it has four interior angles, which are angle CAB, angle ABD, angle DCA, and angle CDB.</note>