Question: As shown in the figure, two lines are intercepted by three parallel lines. AB=2, BC=3, then \frac{EF}{EG} equals ()

Answer: \frac{2}{5}

1. Given AE∥BF∥CG, according to the ##theorem of parallel lines dividing segments proportionally##, we have ##EF/EG=AB/AC##.

2. From ##EF/EG=AB/AC##, it is known that AB=2, BC=3.

3. From AC=AB+BC, we get AC=2+3=5.

4. Substituting the values of AB and AC into the proportion equation, ##EF/EG=2/5##.

5. Through the above reasoning, the final answer is ##2/5##.

平行线; 线段; 内错角

Definition of Parallel Lines

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

Definition of Line Segment

In this problem diagram, <note>line segment AB is a part of a straight line, containing endpoints A and B and all points in between</note>. <note>Line segment BC is a part of a straight line, containing endpoints B and C and all points in between</note>. <note>Line segment AC is a part of a straight line, containing endpoints A and C and all points in between</note>. <note>Line segment EF is a part of a straight line, containing endpoints E and F and all points in between</note>. <note>Line segment EG is a part of a straight line, containing endpoints E and G and all points in between</note>.

Proportional Segments Theorem

If a line is parallel to one side of a triangle and intersects the other two sides or their extensions, then it divides the segments it intersects proportionally in relation to the corresponding segments of the original triangle.

<note>Three parallel lines are AE, BF, and CG</note>, intersecting two lines <note>EG</note> and <note>AC</note>. EF and EG are segments intercepted by these lines, <note>similarly, AB and AC are segments intercepted by these lines</note>. According to the Proportional Segments Theorem, <note>EF/EG=AB/AC</note>.


Question: As shown in the figure, a hotel lobby needs to be carpeted with a circular ring-shaped carpet. The worker only measured the length of the chord AB of the larger circle, which is tangent to the smaller circle, and then calculated the area of the circular ring. If the measured length of AB is 8 meters, then the area of the circular ring is ()

Answer: 16π平方米

1. Draw OC perpendicular to AB through point O, intersecting at point C, and connect OA. According to the perpendicular bisector theorem, OC bisects AB.

2. Therefore, AC equals BC, and AC = AB/2 = 8 meters/2 = 4 meters.

3. According to the problem statement, AB is tangent to the small circle, so OC is the radius of the small circle.

4. Let the radius of the small circle be r, and the radius of the large circle be R. By the Pythagorean theorem, ##OA^2 = OC^2 + AC^2##, i.e., ##R^2 = r^2 + 4^2##.

5. The area of the annulus equals the area of the large circle minus the area of the small circle, i.e., ##πR^2 - πr^2##.

7. Therefore, the area of the annulus is 16π square meters.

Definition of Radius

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

<note>Point O is the center of the circle</note>, <note>Point C is any point on the circle</note>, <note>Line segment OC is the line segment from the center to any point on the circle</note>, therefore <note>Line segment OC is the radius of the small circle</note>; in the large circle, <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 large circle</note>.

Definition of Tangent to a Circle

A line is a tangent to a circle if and only if it has exactly one point of intersection with the circle. This point of intersection is called the point of tangency.

In the figure of this problem, the small circle has only one common point C with <note>line AB</note>. This common point is called the <note>point of tangency</note>. Therefore, <note>line AB is the tangent to the small circle</note>.

Perpendicular Diameter Theorem

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

<note>In the figure of this problem, in circle O, diameter OC is perpendicular to chord AB</note>, then according to the Perpendicular Diameter Theorem, <note>diameter OC bisects chord AB</note>, that is <note>AC=BC</note>, <note>extend OC to intersect the larger circle at D</note>, that is <note>diameter OD bisects the two arcs subtended by chord AB</note>, that is <note>arc AD=arc DB</note>.

Pythagorean Theorem

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

In the diagram of this problem, <note>triangle OAC is a right triangle</note>, in which <note>∠OCA is a right angle (90 degrees)</note>, <note>sides OC and AC are the legs</note>, <note>side OA is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>OA^2 = OC^2 + AC^2</note>, that is <note>R^2 = r^2 + 4^2</note>.

Area Formula of a Circle

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

<note>The radius of the large circle is R</note>, <note>the radius of the small circle is r</note>. According to the area formula of a circle, <note>the area of the large circle A1</note> is equal to pi multiplied by <note>the square of the radius R</note>, which is <note>A1 = πR^2</note>; <note>the area of the small circle A2</note> is equal to pi multiplied by <note>the square of the radius r</note>, which is <note>A2 = πr^2</note>. Therefore, <note>the area of the annulus</note> is equal to <note>the area of the large circle minus the area of the small circle</note>, which is <note>πR^2 - πr^2</note>.


Question: As shown in the figure, in △ABC, points D and E are on sides AB and AC respectively, and DE∥BC. If \frac{AE}{AC}=\frac{3}{4}, AD=9, then AB equals ()

Answer: 12

1. Given that in triangle ABC, points D and E are on sides AB and AC respectively, and DE∥BC.

2. ##According to the parallel line axiom 2 and the similarity theorem (AA)##, in triangles ADE and ABC, since DE∥BC, angle ADE = angle B, and since angle A = angle A, ##triangles ADE and ABC are similar##.

3. By the ##definition## of similar triangles, we obtain the proportion: AD/AB = AE/AC.

4. According to the given condition, AE/AC = 3/4.

5. Substituting the given condition AD = 9, we get the proportion: 9/AB = 3/4.

6. Solving the above proportion, we get AB = 12.

7. Through the above reasoning, the final answer is 12.

平行线; 普通三角形; 位似; 线段

Definition of Similar Triangles

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

<note>In the figure of this problem, triangle ADE and triangle ABC are similar triangles</note>. According to the definition of similar triangles: <note>∠ADE = ∠ABC</note>, <note>∠DEA = ∠BCA</note>, <note>∠EAD = ∠CAB</note>; <note>AD/AB = AE/AC = DE/BC</note>.

Similarity Theorem for Triangles (AA)

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

<note>Angle ADE is equal to angle ABC</note>, and <note>angle A is equal to angle A</note>, so <note>triangle ADE is similar to triangle ABC</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>Two parallel lines DE and BC are intersected by a third line AB</note>, forming the following geometric relationships: Corresponding angles: <note>angle ADE and angle ABC are equal</note>, <note>angle AED and angle ACB are equal</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 DE and BC</note> are intersected by a straight line AB, where <note>angle ADE and angle ABC</note> are on the same side of the transversal AB and on the same side of the intersected lines DE and BC, therefore <note>angle ADE and angle ABC are corresponding angles</note>. Corresponding angles are equal, that is, <note>angle ADE is equal to angle ABC</note>. Similarly, <note>two parallel lines DE and BC</note> are intersected by a straight line AC, where <note>angle AED and angle ACB</note> are on the same side of the transversal AB and on the same side of the intersected lines DE and BC, therefore <note>angle AED and angle ACB are corresponding angles</note>. Corresponding angles are equal, that is, <note>angle AED is equal to angle ACB</note>.


Question: As shown in the figure, Xiaolin walks straight west from point P for 12 meters, then turns left at an angle of α, and walks another 12 meters. He repeats this process, walking a total of 108 meters and returning to point P. Find the value of α - 5.

Answer: 35°

1. It is known that Kobayashi walked straight west from point P for 12 meters, then turned left with an angle α, and walked another 12 meters. He repeated this process and walked a total of 108 meters.

2. Let the number of sides of the polygon formed in this situation be n. The total length of the path is 108 meters, and the length of each side is 12 meters. Therefore, the number of sides can be calculated as n = 108 ÷ 12 = 9.

3. Since Kobayashi turns by an angle α each time, the sum of these angles is the sum of the exterior angles of the polygon. According to the principle of the sum of exterior angles of a polygon, the sum of exterior angles is 360°.

4. An equation can be constructed as n * α = 360°.

5. Substituting the number of sides n = 9, we get 9 * α = 360°.

6. Solving this equation, we get α = 360° ÷ 9 = 40°.

7. According to the problem, we need to find the value of α - 5.

8. Calculation: α - 5 = 40° - 5 = 35°.

9. Therefore, the final answer is 35°.

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>the polygon formed by Xiaolin's walking path has all sides of equal length, which is 12 meters, and all interior angles are equal</note>. Therefore, the polygon formed by Xiaolin's walking path is a <note>regular polygon</note>. <note>It is formed by sequentially connecting multiple line segments, each with a length of 12 meters, totaling 9 line segments</note>. Each pair of line segments intersects only at the endpoints, forming a closed path. <note>Xiaolin starts from point P, walks in the specified direction, and returns to point P after 9 turns, forming a regular 9-sided polygon</note>.

Exterior Angle Sum Theorem of Polygon

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

<note>Xiaolin's rotation angle each time is α</note>, and he rotates 9 times to form a polygon. Therefore, <note>the sum of these 9 exterior angles α is equal to 360°</note>, which means <note>9 * α = 360°</note>.

Definition of Exterior Angle of a Polygon

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

In the figure of this problem, the interior angle of the polygon formed by Xiaolin's walking path is β, and the angle α formed by extending the adjacent side of this interior angle is called the exterior angle of the interior angle β.


Question: As shown in the figure, points B, O, D are on the same line. If ∠1=15°, ∠2=105°, then the degree of ∠AOC is ()

Answer: 90

1. Given ∠2 = 105°, according to the definition of ##adjacent supplementary angles##, ∠BOC and ∠2 are adjacent supplementary angles, therefore ∠BOC = 180° - 105° = 75°.

2. According to the figure in the problem####, we have: ∠AOC = ∠1 + ∠BOC.

3. Substituting the given condition ∠1 = 15° into ####, we have: ∠AOC = 15° + 75° = 90°.

##4##. Through the above reasoning, the final answer is 90°.

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 2 and angle BOC share a common side OC</note>, and their <note>other sides OD and OB are extensions in opposite directions</note>, so <note>angle 2 and BOC are adjacent supplementary angles</note>.


Question: Given that line a∥b, a right-angle triangle plate is placed as shown in the figure. If ∠1=37°, then the degree of ∠2 is ()

Answer: 53°

1. Draw line c parallel to line a, and divide the right angle of the triangle board into ∠3 (near ∠1) and ∠4 (near ∠2). Since line a is parallel to line b, line c is parallel to line a and also parallel to line b.

2. Since line a is parallel to line c, according to the parallel axiom 2 of parallel lines, alternate interior angles are equal, so ∠1 is equal to ∠3.

3. Line c is parallel to line b, according to the parallel axiom 2 of parallel lines, alternate interior angles are equal, so ∠4 is equal to ∠2.

4. According to the problem, ∠3 and ∠4 form the right angle of the triangle board, so their sum is 90°.

6. Given that ∠1 is 37°, therefore ∠2 is equal to 90° minus 37°, which is 53°.

7. Through the above reasoning, the final answer is that the degree of ∠2 is 53°.

平行线; 同位角; 直角三角形

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 b and line c are parallel to line a respectively</note>. According to the transitivity of parallel lines, if <note>line b is parallel to line a</note>, and <note>line c is also parallel to line a</note>, then <note>line b and line c are parallel to each other</note>. Therefore, <note>line b is parallel to line c</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.

Original: <note>Two parallel lines a and c</note> are intersected by <note>the right-angle side of a 60-degree angle of a right triangle ruler</note>, where <note>angle 1 and angle 3 are located between the two parallel lines</note> and on <note>opposite sides of the intersecting line</note>, thus <note>angle 1 and angle 3 are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>angle 1 is equal to angle 3</note>. Similarly, <note>two parallel lines b and c</note> are intersected by <note>the right-angle side of a 30-degree angle of a right triangle ruler</note>, where <note>angle 2 and angle 4 are located between the two parallel lines</note> and on <note>opposite sides of the intersecting line</note>, thus <note>angle 2 and angle 4 are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>angle 2 is equal to angle 4</note>.

Parallel Postulate 2 of Parallel Lines

<note>Line a and line b are intersected by a third line (one side of a right triangle)</note>, forming the following geometric relationships: <note>Alternate interior angles: ∠2 and ∠4 are equal, ∠1 and ∠3 are equal</note>.


Question: As shown in the figure, in the square ABCD with side length 4, first draw an arc with center A and radius AD, then draw an arc with center at the midpoint of AB and radius half of AB. The area of the shaded region between the two arcs is () (result in terms of π).

Answer: 2π

1. According to the problem statement, with point A as the center and the length of AD as the radius, we can draw the sector BAD.

2. Let the midpoint of AB be O, with the midpoint of AB as the center and half the length of AB as the radius, we can draw the semicircle BA.

3. According to the area formula of a sector, (θ/360) * π * R^2, since ABCD is a square, it is evident that θ = 90°, thus the area of sector BAD = πR^2/4. Since R = AD = 4, therefore the area of sector BAD = 4π.

4. According to the area formula of a circle, the area of semicircle BA = πr^2/2. Since r = AB/2 = 2, therefore the area of semicircle BA = 2π.

5. The area of the shaded part is the area of sector BAD minus the area of semicircle BA, i.e., the area of the shaded part = area of sector BAD - area of semicircle BA = 4π - 2π = 2π.

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

正方形; 圆; 弧; 扇形; 中点

Definition of Sector

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

In the <note>sector BAD</note>, <note>the radii AD and AB are two radii of the circle</note>, <note>the arc BD is the arc enclosed by these two radii</note>, so according to the definition of a sector, the figure composed of <note>these two radii and the arc BD enclosed by them</note> is the sector BAD.

Definition of Arc

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

In the figure of this problem, there are two points <note>B and D</note> on the circle, and <note>arc BD</note> is a segment of the curve connecting these two points. According to the definition of an arc, <note>arc BD</note> is a segment of the curve between the two points <note>B and D</note> on the circle. Similarly, there are two points <note>A and B</note> on another circle, and <note>arc AB</note> is a segment of the curve connecting these two points. According to the definition of an arc, <note>arc AB</note> is a segment of the curve between the two points <note>A and B</note> on the circle.

Definition of Semicircle

A semicircle is a geometric figure constructed from a diameter and an arc of a circle, meaning it represents one of the two congruent parts into which a circle is divided by its diameter.

In the figure of this problem, <note>a semicircle is formed by the diameter AB and the arc BA</note>. Using <note>the midpoint of AB as the center</note>, and <note>half the length of AB as the radius to draw the arc</note>, the figure formed by this arc is a semicircle, where <note>the radius is half of AB (i.e., 2)</note> and <note>the center is the midpoint of AB</note>.

Definition of Square

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

In the figure of this problem, in quadrilateral ABCD, <note>sides AB, BC, CD, and DA are equal</note>, and <note>angles DAB, ABC, BCD, and CDA are all right angles (90 degrees)</note>, so <note>ABCD is a square</note>.

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 central angle BAD of the sector BAD is 90 degrees</note>, <note>the length of the radius AD is 4</note>. According to the formula for the area of a sector, the area A of the sector can be calculated using the formula A = (θ/360) * π * ##r^2##, where θ is the degree measure of the central angle and r is the length of the radius. So, <note>the area A of the sector BAD = (90/360) * π * ##4^2##</note>.

Area Formula of a Circle

In <note>circle O</note>, <note>the radius of the circle is AO</note>. According to the area formula of a circle, <note>the area of the circle A is equal to pi π multiplied by the square of the radius AO</note>, i.e., A = π x AO².

Definition of Radius

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

<note>In the diagram of this problem, in circle A, point A is the center of the circle</note>, <note>point D is any point on the circle</note>, <note>line segment AD is the line segment from the center to any point on the circle</note>, therefore <note>line segment AD is the radius of the circle</note>. Similarly, <note>in circle O, point O is the center of the circle</note>, <note>point A is any point on the circle</note>, <note>line segment AO is the line segment from the center to any point on the circle</note>, therefore <note>line segment AO is the radius of the circle</note>.

Midpoint of a Line Segment

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

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


Question: As shown in the figure, in parallelogram ABCD, F is a point on AD, CF=CD. If ∠B=72°, then the degree of ∠AFC is ()

Answer: 108°

1. Given quadrilateral ABCD is a parallelogram, according to the properties of parallelograms, the opposite angles of a parallelogram are equal, thus ∠D = ∠B = 72°.

2. Given CF = CD, according to the properties of isosceles triangles, if two sides of a triangle are equal, then the angles opposite these sides are equal, thus ∠DFC = ∠D = 72°.

3. Since ∠DFC + ∠AFC = 180° (the sum of adjacent angles on the same straight line is 180°), therefore ∠AFC = 180° - ∠DFC = 180° - 72° = 108°.

4. Through the above reasoning, the final answer is ∠AFC = 108°.

平行四边形; 等腰三角形; 邻补角

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

Definition of Isosceles Triangle

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

<note>In triangle FCD, side CF and side CD are equal</note>, therefore <note>triangle FCD is an isosceles triangle</note>.

Properties of Parallelogram Theorem

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

In the figure of this problem, in parallelogram ABCD, <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, the intersection point divides diagonal AC into two equal segments <note>AO and OC</note>, and divides diagonal BD into two equal segments <note>BO and OD</note>. Therefore, <note>∠B=∠D=72°</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 CFD, side CF and side CD are equal</note>. Therefore, according to the properties of the isosceles triangle, <note>the angles opposite the equal sides are equal, i.e., ∠DCF = ∠DFC</note>.

Definition of Linear Pair of Angles

<note>Angle AFC and angle DFC share a common side FC</note>, their <note>other sides FA and FD are extensions in opposite directions</note>, so <note>angle AFC and angle DFC are adjacent supplementary angles</note>.


Question: As shown in the figure, two right-angled triangles are overlapped with their right-angle vertices coinciding. If ∠1=40°, then the measure of ∠2 is ()

Answer: 40°

1. According to the problem statement, when the right-angle vertices of the two triangular plates coincide and overlap, there are two right angles in the figure.

2. Let the angle between ∠1 and ∠2 be ∠3.

3. According to the problem statement, ∠1 = 40°.

4. Based on the given conditions, it can be deduced that ∠1 + ∠3 = 90° (because the two angles are adjacent to the right-angle vertex).

6. Similarly, ∠2 + ∠3 = 90° (the other adjacent right-angle vertex).

7. Substituting the value of ∠3, we get ∠2 + 50° = 90°.

8. Solving this equation, we get ∠2 = 90° - 50° = 40°.

9. Based on the above reasoning, the final answer is ∠2 = 40°.

Definition of Right Triangle

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

<note>In the figure of this problem, in the two right triangles</note>, <note>each right triangle has one right angle (90 degrees)</note>, therefore they are both right triangles. Each right triangle's <note>right angle is composed of two acute angles</note>, <note>one is the sum of ∠1 and ∠3</note>, the other is <note>the sum of ∠2 and ∠3</note>.


Question: As shown in the figure, in △ABC, DE∥BC, \frac{AD}{DB}=\frac{1}{2}, DE=4, then the length of BC is ()

Answer: 12

2. Therefore, \frac{AD}{AB} = \frac{1}{3}.

3. In △ABC, DE∥BC, so according to the theorem of proportional segments cut by parallel lines and the similarity theorem of triangles (SAS), we get △ADE∽△ABC, thus \frac{DE}{BC} = \frac{AD}{AB}.

4. In summary, \frac{DE}{BC} = \frac{1}{3}.

5. Given DE=4, therefore BC = 3, so DE = 3 \times 4 = 12.

6. Through the above reasoning, the final length of BC is 12.

普通三角形; 平行线; 线段; 位似

Proportional Segments Theorem

If a line is parallel to one side of a triangle and intersects the other two sides or their extensions, then it divides the segments it intersects proportionally in relation to the corresponding segments of the original triangle.

<note>In the figure of this problem, in triangle ABC, line DE is parallel to side BC and intersects the other two sides AB and AC at points D and E</note>, according to the Proportional Segments Theorem, we have: <note>AD/DB = AE/EC</note>. Specifically, in this problem, <note>AD/DB = 1/2, DE/BC = 1/3</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>Triangles ADE and ABC are similar triangles</note>. According to the definition of similar triangles: <note>angle ADE = angle ABC</note>, <note>angle AED = angle ACB</note>, <note>angle A = angle A</note>; <note>AD/AB = DE/BC = AE/AC</note>.

SAS Criterion for Similar Triangles

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

<note>Side AD corresponds to side AB</note>, <note>side AE corresponds to side AC</note>, and <note>side AD/side AB = side AE/side AC</note>, and <note>angle A = angle A</note>, so according to the Side-Angle-Side similarity criterion (SAS criterion), <note>triangle ADE is similar to triangle ABC</note>.


Question: As shown in the figure, DE∥BC, AD=3, DB=4, AE=1.5, then EC equals ()

Answer: 2

1. Given DE∥BC, ##according to the parallel axiom 2 of parallel lines, ∠AED＝∠ACB, ∠ADE＝∠ABC, according to the theorem of determining similar triangles (AA)##, we can determine △ADE∽ △ABC.

2. According to ##the definition of similar triangles##, for △ADE and △ABC, we have: ##AD/AB = AE/AC = DE/BC##.

3. Combining the given conditions AD=3, DB=4, therefore AB=AD+DB=3+4=7.

4. Continuing to combine the given conditions AE=1.5, substituting the known proportional relationship into the corresponding sides of the similar triangles####, we can obtain: ##AD/DB = AE/EC##.

5. Substituting the known values into the proportional relationship in step 4: ##3/4 = 1.5/EC##.

6. Solving the proportional equation ##3/4 = 1.5/EC##, we can obtain EC = 2.

7. Through the above reasoning, the final answer is 2.

平行线; 内错角; 普通三角形

Definition of Similar Triangles

<note>Triangle ADE and triangle ABC are similar triangles</note>. According to the definition of similar triangles: <note>∠ADE = ∠ABC</note>, <note>∠DEA = ∠BCA</note>, <note>∠EAD = ∠CAB</note>; <note>AD/AB = AE/AC = DE/BC</note>. Specifically, <note>AD corresponds to AB</note>, <note>AE corresponds to AC</note>, <note>DE corresponds to BC</note>.

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>AED</note> and triangle <note>ABC</note>, if <note>angle AED is equal to angle ACB</note>, and <note>angle ADE is equal to angle ABC</note>, then triangle <note>AED is similar to triangle ABC</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>Two parallel lines ED and CB</note> are intersected by a third line <note>AC</note>, forming the following geometric relationship: corresponding angles: <note>angle AED and angle ACB are equal</note>. Two parallel lines <note>ED and CB</note> are intersected by a third line <note>AB</note>, forming the following geometric relationship: corresponding angles: <note>angle ADE and angle ABC are equal</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 ED and CB are intersected by a line AC</note>, <note>where angle AED and angle ACB are on the same side of the intersecting line AC and on the same side of the intersected lines ED and CB</note>, therefore <note>angle AED and angle ACB are corresponding angles</note>. <note>Corresponding angles are equal</note>, that is <note>angle AED is equal to angle ACB</note>. <note>Two parallel lines ED and CB are intersected by a line AB</note>, <note>where angle ADE and angle ABC are on the same side of the intersecting line AB and on the same side of the intersected lines ED and CB</note>, therefore <note>angle ADE and angle ABC are corresponding angles</note>. <note>Corresponding angles are equal</note>, that is <note>angle ADE is equal to angle ABC</note>.


Question: As shown in the figure, if △ABC∽△ACD, ∠A=60°, ∠ACD=40°, then the measure of ∠BCD is ()

Answer: 40°

1. According to the given conditions, △ABC∽△ACD, and it is also known that ∠A=60°, ∠ACD=40°.

2. Based on the property that corresponding angles of similar triangles are equal, we get ∠B = ∠ACD = 40°.

3. Using the interior angle sum theorem of triangles (i.e., the sum of the three interior angles of a triangle is 180°), in △ABC, we get ∠ACB = 180° - ∠A - ∠B = 180° - 60° - 40° = 80°.

4. In △BCD, through the relationship between angle ACB and angle ACD, we can get ∠BCD = ∠ACB - ∠ACD = 80° - 40° = 40°.

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

普通三角形; 三角形的外角; 对顶角

Definition of Similar Triangles

<note>Triangle ABC and triangle ACD are similar triangles</note>. According to the definition of similar triangles: <note>∠BAC = ∠DAC</note>, <note>∠ABC = ∠ACD</note>, <note>∠BCA = ∠CDA</note>; <note>AB/AC = AC/AD = BC/CD</note>.

Triangle Angle Sum Theorem

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

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


Question: As shown in the figure, a car goes through a section of road and makes two turns, after which it is traveling in the same direction as it was originally, meaning the two sections of road before and after the turns are parallel to each other. The angle of the first turn ∠B is equal to 142°, the degree of the second turn ∠C is ()

Answer: 142°

1. Given that the two roads after two turns are parallel to each other, i.e., AB is parallel to CD.

2. The angle of the first turn, ##∠ABC##, is 142°.

3. Since AB is parallel to CD, and ∠ABC and the unknown angle ##∠DCB## are in alternate interior positions between the parallel lines, according to the Parallel Postulate 2 theorem, we get ##∠DCB=∠ABC (alternate interior angles are equal)##.

4. From the above reasoning, we get ##∠DCB##=142°.

5. Through the above reasoning, the final answer is 142°.

邻补角; 平行线; 同位角

Definition of Parallel Lines

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

In the diagram of this problem, <note>line AB and line CD</note> are located in the same plane, and <note>they have no intersection points</note>, so according to the definition of parallel lines, <note>line AB and line CD are parallel lines</note>.

Parallel Postulate 2 of Parallel Lines

<note>AB is parallel to CD</note>, therefore the alternate interior angles formed by the <note>transversal BC</note> between <note>AB and CD</note> are equal, which means the <note>measure of ∠ABC is equal to the measure of ∠DCB</note>. Hence, <note>∠DCB</note>=∠ABC=142°.

Definition of Alternate Interior Angles

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

<note>Two parallel lines AB and CD</note> are intersected by <note>a transversal xx</note>, where <note>angle ABC and angle DCB are located between the two parallel lines and on opposite sides of the transversal BC</note>, thus <note>angle ABC and angle DCB are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>angle ABC is equal to angle DCB</note>.


Question: As shown in the figure, DE is the midline of △ABC. Given that the area of △ABC is 8cm^{2}, find the area of △ADE in cm^{2}.

Answer: 2

1. We know DE is the midline of △ABC, indicating D and E are the midpoints of AB and AC respectively.

2. According to the midline theorem of triangles, DE ∥ BC and DE is half of BC.

3. Since D and E are the midpoints of AB and AC, according to the ##similar triangles criterion (SAS)##, △ADE is similar to △ABC.

4. Because DE is half of BC, the ratio is 1:2, thus the similarity ratio is 1:2.

5. The area ratio of similar triangles equals the square of the similarity ratio, so the area ratio of △ADE and △ABC is (1/2)^2, which is 1:4.

6. Given the area of △ABC is 8 ##cm^2##, the area of △ADE is 8 ##cm^2## divided by 4, which equals 2 ##cm^2##.

7. Through the above reasoning, the final answer is 2 ##cm^2##.

中点; 普通三角形; 平行线

Definition of Similar Triangles

<note>△ADE and △ABC are similar triangles</note>. According to the definition of similar triangles: <note>∠ADE = ∠ABC</note>, <note>∠DEA = ∠BCA</note>, <note>∠EAD = ∠CAB</note>; <note>AD/AB = AE/AC = DE/BC = 1:2</note>.

Triangle Midline Theorem

In a triangle, if a line segment connects the midpoints of two sides, then this line segment is parallel to the third side and is equal to half the length of the third side.

<note>In triangle ABC, point D is the midpoint of side AB, and point E is the midpoint of side AC, segment DE connects these two midpoints</note>. According to the Triangle Midline Theorem, <note>segment DE is parallel to the third side BC and is equal to half of the third side BC</note>, i.e., <note>DE ∥ BC</note>, and <note>DE = 1/2 * BC</note>.

Theorem on the Area Ratio of Similar Triangles

If two triangles are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding sides.

<note>Triangle ADE and triangle ABC are similar triangles</note>, <note>the ratio of side DE to side BC is 1:2</note>, that is, <note>the similarity ratio is 1:2</note>. Therefore, <note>the ratio of the area of triangle ADE to the area of triangle ABC is equal to (1/2)^2 = 1:4</note>.

SAS Criterion for Similar Triangles

In the <note>triangles ADE and ABC</note>, <note>side AD corresponds to side AB</note>, <note>side AE corresponds to side AC</note>, and <note>side AD/side AB = side AE/side AC</note>, and <note>angle DAE = angle BAC</note>. Therefore, according to the Side-Angle-Side similarity criterion (SAS criterion), <note>triangle ADE is similar to triangle ABC</note>.


Question: As shown in the figure, line AB intersects line CD at point E. On the bisector of ∠CEB, there is a point F, and FM∥AB. When ∠3=10°, the degree of ∠F is ()

Answer: 85°

1. Given that line AB intersects line CD at point E, there is a point F on the bisector of ∠CEB, and FM is parallel to line AB.

2. According to the definition of a straight angle, ∠CED can be expressed as ∠1 + ∠2 + ∠3. Since ∠3 = 10°, we can obtain ∠1 + ∠2 = 180° - ∠3 = 180° - 10° = 170°.

3. Since EF bisects ∠CEB, according to the definition of an angle bisector, we can conclude that ∠1 = ∠2.

4. From ∠1 + ∠2 = 170° and ∠1 = ∠2, we can derive that ∠1 = ∠2 = 170° / 2 = 85°.

5. According to the problem statement, FM is parallel to line AB. Therefore, ∠F and ∠2 are alternate interior angles of parallel lines. According to the theorem of alternate interior angles of parallel lines being equal, we get ∠F = ∠2.

6. In summary, ∠F = 85°.

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

平行线; 对顶角; 等腰三角形; 内错角; 邻补角

Definition of Angle Bisector

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

<note>The vertex of angle CEB is point E</note>, <note>a line EF is drawn from point E</note>, this line <note>divides angle CEB into two equal angles</note>, namely <note>angle CEF (∠1) and angle BEF (∠2) are equal</note>. Therefore, <note>line EF is the angle bisector of angle CEB</note>.

Parallel Postulate 2 of Parallel Lines

In the diagram of this problem, two parallel lines <note>FM</note> and <note>AB</note> are intersected by a third line <note>EF</note>, forming the following geometric relationships: 1. Corresponding angles: ##none##. 2. Alternate interior angles: <note>angle F</note> and <note>angle 2</note> are equal. 3. Same-side interior angles: <note>angle AEF</note> and <note>angle F</note> are supplementary, that is, <note>angle AEF + angle F = 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 same-side interior angles are supplementary.

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>Ray C rotates around endpoint E to form a straight line with the initial side, creating straight angle CED</note>. According to the definition of a straight angle, <note>a straight angle measures 180 degrees, i.e., angle CED = 180 degrees</note>.


Question: As shown in the figure, AB=AC, AD=AE, ∠BAC=∠DAE, ∠1=25°, ∠2=30°, then ∠3=()

Answer: 55°

1. Given AB=AC, AD=AE, and ∠BAC=∠DAE.

2. ∠BAC-∠DAC=∠DAE-∠DAC, thus ∠BAD=∠EAC.

3. In △BAD and △EAC, we have AB=AC, ∠BAD=∠EAC, AD=AE.

4. According to the SAS congruence condition, △BAD≌△CAE (SAS), then the corresponding angles ∠ABD=∠ECA=30°.

5. Given ∠ABD=30°, ∠1=25°.

6. In △ABD, according to the exterior angle theorem, ∠3=∠1+∠ABD.

7. ∠3=25°+30°=55°.

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

等腰三角形; 普通三角形; 对顶角; 三角形的外角

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 ABD, <note>angle 3 is an exterior angle of the triangle</note>, <note>angle 1 and angle ABD are the two non-adjacent interior angles to exterior angle 3</note>, according to the Exterior Angle Theorem of Triangle, <note>exterior angle 3 is equal to the sum of the two non-adjacent interior angles 1 and angle ABD</note>, that is, <note>angle 3 = angle 1 + angle ABD</note>.

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, <note>an interior angle of polygon ABD is ∠ADB</note>, and the angle formed by extending the adjacent sides <note>BD and DA to form angle ∠3</note> is called the exterior angle of the interior angle <note>∠ADB</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.

In the figure of this problem, in triangles BAD and EAC, <note>side AB is equal to side AC</note>, <note>side AD is equal to side AE</note>, and <note>the included angle ∠BAD is equal to the included angle ∠EAC</note>. Therefore, according to the <note>Triangular Congruence Theorem (SAS)</note>, these two triangles are congruent.

Definition of Congruent Triangles

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

<note>Triangles BAD and EAC are congruent triangles</note>, and the <note>corresponding sides and corresponding angles</note> of triangle BAD are equal to those of triangle EAC, namely:
<note>side BA = side AC</note>
<note>side BD = side EC</note>
<note>side AD = side AE</note>
At the same time, <note>the corresponding angles are also equal</note>:
<note>angle 1 = angle EAC</note>
<note>angle ABD = angle ACE</note>
<note>angle ADB = angle AEC</note>


Question: As shown in the figure, there is a pond. To measure the distance between the two ends A and B of the pond, first select a point C on the ground that can directly reach points A and B without passing through the pond. Connect AC and extend it to D, making CD = CA. Connect BC and extend it to E, making CE = CB. Connect ED. If DE is measured to be 58 meters, then the distance between A and B is ()

Answer: 58米

1. Draw a straight line from point A to point C and extend it to point D, such that CD=CA.

2. Draw a straight line from point B to point C and extend it to point E, such that CE=CB.

3. Measure DE=58 meters using a measuring instrument.

4. In △ABC and △DEC, it is known that AC=CD, ##CE=CB##.

####

##5##. ##According to the definition of vertical angles##, it follows that ##∠ACB##=∠DCE.

##6##. According to ##the congruent triangles theorem (SAS)##, in triangles ABC and DEC, side AC is equal to side CD, ##∠ACB## is equal to ∠DCE, and side CE is equal to side CB, hence △ABC##≌##△DEC.

##7##. According to ##the definition of congruent triangles##, it follows that AB=DE.

##8##. From the above conclusion and the known DE=58 meters, it follows that AB=58 meters.

Definition of Vertical Angles

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

<note>Line AD</note> and <note>BE</note> intersect at point <note>C</note>, forming four angles: <note>∠ACB</note>, <note>∠BCD</note>, <note>∠ACE</note>, and <note>∠ECD</note>. According to the definition of vertical angles, <note>∠ACB</note> and <note>∠ECD</note> are vertical angles, and <note>∠BCD</note> and <note>∠ACE</note> are vertical angles. Since the angles of vertical angles are equal, <note>∠BCD=∠ACE</note>, <note>∠ACB=∠ECD</note>.

Triangular Congruence Theorem (SAS)

In <note>triangle ACB</note> and <note>triangle DCE</note>, <note>side AC is equal to side CD</note>, <note>side CB is equal to side CE</note>, and <note>angle ACB is equal to angle DCE</note>. Therefore, according to the Triangular Congruence Theorem (SAS), these two triangles are congruent.

Definition of Congruent Triangles

In the figure of this problem, <note>triangle ACB and triangle DCE are congruent triangles</note>, <note>the corresponding sides and corresponding angles of triangle ACB are equal to those of triangle DCE</note>, namely: <note>side AB = side ED</note> <note>side AC = side DC</note> <note>side BC = side EC</note>, and <note>the corresponding angles are also equal</note>: <note>angle ACB = angle DCE</note> <note>angle CAB = angle CDB</note> <note>angle CBA = angle CED</note>.


Question: Given as shown in the figure, in ⊙O, OA⊥OB, ∠A=35°, then the degree of arc CD is ()

Answer: 20°

1. Connect OC. Given that OA is perpendicular to OB, according to the definition of perpendicular lines, we get ∠AOB=90°.

2. Given ∠A=35°, since OB=OC, triangle OBC is an isosceles triangle, which means ∠OBC=∠OCB. According to the sum of the interior angles of a triangle being 180 degrees, we get ∠OBC=∠OCB=55°.

####

##3##. From ∠OBC=∠OCB=55°, according to the triangle interior angle sum theorem, we get ∠COB=70°.

##4. Because ∠AOB=∠COB+∠COD=90°, and ∠COB=70°, so ∠COD=∠AOB-∠COB=90°-70°=20°.

##5. According to the degree of the arc being the degree of the central angle it subtends, in the circle, the degree of chord CD is the degree of the corresponding central angle COD, and finally we get the degree of arc CD as 20°##.

圆; 垂线; 圆周角; 弧; 直角三角形

Definition of Perpendicular Lines

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

<note>Line OA and line OB intersect to form a 90-degree angle ∠AOB</note>, therefore according to the definition of perpendicular lines, <note>line OA and line OB are perpendicular to each other</note>.

Definition of Isosceles Triangle

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

<note>Side OB and side OC are equal</note>, therefore <note>triangle OBC is an isosceles triangle</note>.

Triangle Angle Sum Theorem

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

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


Question: As shown in the figure, in parallelogram ABCD, ∠C=120°, AB=AE=5, AE intersects BD at point F, and AF=2EF. Find the length of BC.

Answer: 10

1. Given that quadrilateral ABCD is a parallelogram, according to the properties of parallelograms, AD = BC, and AD ∥ BC.

2. Since AB = AE = 5, AE intersects BD at point F, and AF = 2EF.

3. Suppose we draw auxiliary lines, connecting EF and DF.

4. In △AFD and △EFB, we have ∠AFD = ∠EFB (vertical angles are equal), and ∠ADF = ∠EBF (alternate interior angles between parallel lines are equal), so according to the AA similarity theorem, △AFD ∽ △EFB.

5. Therefore, for the similar triangles △AFD and △EFB, we have rac{BE}{AD} = rac{EF}{AF}.

6. Given AF = 2EF, so rac{EF}{AF} = rac{1}{2}.

7. Therefore, AD = 2BE.

8. Since ∠C = 120 degrees, according to the property of supplementary angles in parallelograms, ∠ABC = 60 degrees. In triangle ABE, since AB = AE = 5, ∠ABE = 60 degrees, according to the theorem of equilateral triangles (60-degree angle in an isosceles triangle), we know triangle ABE is an equilateral triangle, so BE = 5.

9. Therefore, AD = 2 * 5 = 10.

10. According to the properties of parallelograms, AD = BC, so BC = 10.

11. Through the above reasoning, the final answer is that the length of BC is 10.

平行四边形; 线段; 等腰三角形; 三角形的外角; 内错角

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

Definition of Vertical Angles

<note>Line AF and line BD intersect at point F</note>, forming four angles: <note>∠AFD, ∠EFB, ∠AFE, and ∠DFB</note>. According to the definition of vertical angles, <note>∠AFD and ∠EFB are vertical angles</note>, <note>∠AFE and ∠DFB are vertical angles</note>. Since vertical angles are equal, <note>∠AFD = ∠EFB</note>, <note>∠AFE = ∠DFB</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>Two parallel lines AD and BC</note> are intersected by <note>a third line BD</note>, forming the following geometric relationship: <note>Alternate interior angles: angle ADF and angle FBE</note> are equal.

In a parallelogram, each pair of adjacent interior angles are supplementary, meaning their sum is 180°.

In the figure of this problem, in parallelogram ABCD, angle C and angle ABC are adjacent interior angles. According to the Adjacent Angles Supplementary Theorem of Parallelogram, <note>angle C + angle ABC = 180°.</note>

Similarity Theorem for Triangles (AA)

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

<note>In triangles AFD and EFB</note>, ∠AFD=∠EFB are <note>equal opposite angles</note>, ∠ADF=∠EBF are <note>equal alternate interior angles</note>, therefore <note>triangle AFD is similar to triangle EFB</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>In the figure of this problem, triangle AFD and triangle EFB are similar triangles</note>. According to the definition of similar triangles: <note>∠AFD = ∠EFB (vertical angles)</note>, <note>∠ADF = ∠EBF (alternate interior angles between parallel lines)</note>, <note>∠DAF = ∠FEB (alternate interior angles between parallel lines)</note>; <note>AF/EF = DF/FB = AD/BE</note>. Given <note>AF=2EF</note>, so <note>EF/AF=1/2</note>.

Equilateral Triangle Identification Theorem (60-Degree Angle in an Isosceles Triangle)

An isosceles triangle with one interior angle measuring 60 degrees is an equilateral triangle.

In the figure of this problem, it is known that <note>△ABE is an isosceles triangle, with sides AB and AE equal</note>, and there is an interior angle of 60°, i.e., <note>∠ABE=60°</note>. According to the Equilateral Triangle Identification Theorem, if an isosceles triangle has an interior angle of 60°, then the lengths of its three sides are equal, and all three interior angles are 60°. Therefore, it can be determined that <note>△ABE is an equilateral triangle</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 ABE is an equilateral triangle</note>. <note>Sides AB, AE, and BE are of equal length</note>, and <note>angles ABE, AEB, and EAB are equal in measure, each being 60°</note>.

Definition of Isosceles Triangle

<note>Side AB and side AE are equal</note>, therefore triangle ABE is an isosceles triangle.

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.

In the diagram of this problem, <note>two parallel lines AD and BC</note> are intersected by <note>a line BD</note>, where <note>angle ADF</note> and <note>angle FBE</note> are located <note>between the two parallel lines</note> and on <note>opposite sides of the intersecting line BD</note>. Therefore, <note>angle ADF and angle FBE</note> are alternate interior angles. Alternate interior angles are equal, that is, <note>angle ADF</note> is equal to <note>angle FBE</note>.


Question: As shown in the figure, in the isosceles triangle ABC, AB=AC, BD is the altitude on side AC. If ∠A=36°, then the measure of ∠DBC is ()

Answer: 18°

1. Given that in the isosceles triangle ABC, AB=AC, ∠A=36°, according to the properties of isosceles triangles, we get ∠ABC=∠ACB.

2. Since the sum of the interior angles of a triangle is 180°, we get ∠ABC + ∠ACB + ∠A = 180°.

3. Substituting the given conditions, 36° + 2∠ABC = 180°, i.e., 2∠ABC = 144°, we get ∠ABC = 72°.

4. From ∠ABC = ∠ACB = 72°, we can deduce that ∠ACB = 72°.

5. Since BD is the height on side AC, according to the definition, we get BD⊥AC.

6. Since BD is the height on side AC, ∴∠BDA = 90°.

7. From ∠ACB = 72°, in the right triangle BDC, we get ∠DBC = ∠BDA - ∠ACB.

8. Substituting the given conditions, ∠DBC = 90° - 72° = 18°.

9. Through the above reasoning, we finally get the answer as 18°.

等腰三角形; 垂线; 直角三角形; 三角形的外角

Definition of Isosceles Triangle

The sides <note>AB</note> and <note>AC</note> are equal, therefore <note>triangle ABC is an isosceles triangle</note>.

Exterior Angle Theorem of Triangle

In <note>triangle BDC</note>, <note>angle BDA is an exterior angle of the triangle</note>, <note>angle DBC and angle ACB are the two non-adjacent interior angles to the exterior angle BDA</note>. According to the Exterior Angle Theorem of Triangle, <note>the exterior angle BDA is equal to the sum of the two non-adjacent interior angles DBC and ACB</note>, that is, <note>angle BDA = angle DBC + angle ACB</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>The line segment BD perpendicular to the opposite side AC from vertex B is the altitude of vertex B</note>.<note>The line segment BD forms a right angle (90 degrees) with side AC</note>, which indicates that the line segment BD is the perpendicular distance from vertex B to the opposite side AC.

Properties of Isosceles Triangle

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

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

Triangle Angle Sum Theorem

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

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


Question: As shown in the figure, AD is the median of △ABC. Given that the perimeter of △ABD is 22cm and AB is 3cm longer than AC, find the perimeter of △ACD.

Answer: 19cm

1. From the given conditions, AB is 3 cm longer than AC, i.e., AB = AC + 3 cm.

2. Since AD is the median of △ABC, ##according to the definition of the triangle median,## we have BD = DC.

3. ##According to the formula for the perimeter of a triangle, given that## the perimeter of △ABD is 22 cm, the sum of all sides of ABD is 22 cm, i.e., AB + BD + AD = 22 cm.

4. Replacing AB with AC, we get AC + 3 + ##BD## + AD = 22 cm.

5. Combining with step 2, we have DC = BD, thus the equation becomes AC + 3 + ##DC## + AD = 22 cm.

6. ##According to the formula for the perimeter of a triangle, we know the perimeter of △ACD = AD + AC + DC, substituting into step 5 we get AD + AC + DC = 22 cm - 3.##

####

##7##. Therefore, the perimeter of △ACD is AC + DC + AD = 19 cm.

普通三角形; 线段; 中点

Definition of Median of a Triangle

A median of a triangle is a line segment drawn from one vertex of the triangle to the midpoint of the opposite side.

In <note>triangle ABC</note>, <note>vertex A</note> is one of the vertices of the triangle, and the opposite side <note>BC</note> is the side opposite to vertex A. <note>Point D is the midpoint of side BC</note>, <note>line segment AD is the segment from vertex A to midpoint D of opposite side BC</note>, therefore <note>AD is a median of triangle ABC</note>.

Formula for the Perimeter of a Triangle

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

In the figure of this problem, it is known that the three sides of triangle <note>ABD</note> are <note>AB</note>, <note>AD</note>, and <note>BD</note>. According to the formula for the perimeter of a triangle, the perimeter of the triangle is equal to the sum of the lengths of its three sides, that is, the perimeter <note>P=AB+AD+BD</note>. Similarly, the three sides of triangle <note>ACD</note> are <note>AC</note>, <note>AD</note>, and <note>CD</note>. According to the formula for the perimeter of a triangle, the perimeter of the triangle is equal to the sum of the lengths of its three sides, that is, the perimeter <note>P=AC+AD+CD</note>.


Question: As shown in the figure, if CB=4cm, DB=7cm, and D is the midpoint of AC, then the length of AC is ()

Answer: 6cm

2. ####According to segment subtraction, we get CD=7cm-4cm=3cm.

3. From the given condition that D is the midpoint of AC, according to the definition of the midpoint of a segment, AD=CD.

5. According to segment addition, AC=AD+DC.

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

Midpoint of a Line Segment

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

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


Question: As shown in the figure, DE is the perpendicular bisector of side BC of △ABC, intersecting BC at E and intersecting AB at D. Given ∠B=40°, ∠A=60°, find the degree measure of ∠ACD.

Answer: 40°

1. Given ∠B = 40°, ∠A = 60°. In △ABC, by the triangle angle sum theorem, we can obtain ∠ACB = 180° - 60° - 40° = 80°.

2. DE is the perpendicular bisector of side BC of △ABC. According to the properties of the perpendicular bisector, ##BD＝DC##, ∠BCD = ∠B = 40°.

3. In △ACD, from the previous step we can get ∠ACD = ∠ACB - ∠BCD = 80° - 40° = 40°.

4. Through the above reasoning, the final answer is ∠ACD = 40°.

垂直平分线; 等腰三角形; 直角三角形

Definition of Perpendicular Bisector

A line is a perpendicular bisector if and only if it passes through the midpoint of a line segment and is perpendicular to that line segment.

In the figure of this problem, <note>line DE passes through the midpoint E of segment BC</note>, and <note>line DE is perpendicular to segment BC</note>. Therefore, <note>line DE is the perpendicular bisector of segment BC</note>. According to the definition of the perpendicular bisector, <note>E is the midpoint of BC, that is, BE = EC</note>, and <note>DE is perpendicular to BC, that is, ∠DEB = ∠DEC = 90°</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>In the diagram of this problem, ∠B, ∠A, and ∠ACD are all examples of angles.</note> For example, <note>∠B is a geometric figure formed by ray BA and ray BC, which share a common endpoint B. This common endpoint B is called the vertex of ∠B, and ray BA and ray BC are called the sides of ∠B.</note> <note>∠A is a geometric figure formed by ray AB and ray AC, which share a common endpoint A. This common endpoint A is called the vertex of ∠A, and ray AB and ray AC are called the sides of ∠A.</note> <note>∠ACD is a geometric figure formed by ray AC and ray DC, which share a common endpoint C. This common endpoint C is called the vertex of ∠ACD, and ray AC and ray DC are called the sides of ∠ACD.</note>

Triangle Angle Sum Theorem

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

Properties of the Perpendicular Bisector

Any point on the perpendicular bisector of a line segment is equidistant from the endpoints of the segment.

Original: <note>DE is the perpendicular bisector of side BC of △ABC</note>. According to the properties of the perpendicular bisector, <note>any point on DE is equidistant from the endpoints B and C of segment BC</note>, that is, <note>BE = CE</note> and <note>BD = DC</note>.


Question: As shown in the figure, in the circle ⊙O with a radius of 5cm, AB is a chord, OC⊥AB at point C, and OC=3cm. Find the length of AB.

Answer: 8cm

1. Connect OA, knowing the radius OA=5cm.

2. Since OC is perpendicular to AB at point C, ##according to the diameter theorem##, point C is the midpoint of AB. Therefore, we have: AB=2AC.

3. In the right triangle △OAC, ∠ACO=90°. According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs, that is: ##OA^2 = AC^2 + OC^2##.

4. Since OC=3cm and OA=5cm, substitute into the above formula: ##5^2 = AC^2 + 3^2##.

5. Solving the equation, we get: 25 = ##AC^2## + 9, that is ##AC^2## = 16, therefore AC = √16 = 4cm.

6. From step 2, we get: AB = 2AC = 2 * 4 = 8cm.

7. Through the above reasoning, the final answer is 8cm.

圆; 弦; 垂线; 直角三角形; 线段

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>.

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>, <note>line segment OC is perpendicular to chord AB</note>, then according to the <note>Perpendicular Diameter Theorem</note>, <note>line segment OC bisects chord AB</note>, that is, <note>AC=BC</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 OAC, <note>angle ACO is a right angle (90 degrees)</note>, therefore <note>triangle OAC is a right triangle</note>. <note>Side CA</note> and <note>side OC are the legs</note>, <note>side AO 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.

In the figure of this problem, in the right triangle △OAC, <note>angle ACO is a right angle (90 degrees)</note>, sides OC and AC are the legs, and side OA is the hypotenuse, so according to the Pythagorean Theorem, <note>OA^2 = OC^2 + AC^2</note>.


Question: As shown in the figure, in Rt△ABC, ∠BAC=90°, △ABC is rotated clockwise by 90° around point A to obtain △AB′C′ (the corresponding point of point B is point B′, the corresponding point of point C is point C′). Connect CC′, if ∠CC′B′=33°, then the size of ∠B is ()

Answer: 78°

1. Given that in the right triangle ABC, ∠BAC=90°. After rotating △ABC clockwise by 90° around point A, we obtain △AB'C'.

2. According to the properties of rotation, AC = AC', and ####∠CAC'=90°.

3. According to the properties of rotation, the shape remains unchanged after rotation, so we can conclude ∠AB'C' = ∠B.

##4.## Connect CC', in △ACC', since AC = AC' and ∠CAC'=90°, we have ∠ACC'=45°.

##5.## Given ∠CC'B' = 33°, according to ##the exterior angle theorem##, ∠AB'C' = ∠ACC' + ∠CC'B'.

直角三角形; 旋转; 对顶角; 邻补角; 平行线

Definition of Right Triangle

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

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 ACC', side AC and side AC' are equal</note>, therefore <note>triangle ACC' is an isosceles triangle</note>.

Properties of Isosceles Triangle

In the figure of this problem, in the isosceles triangle <note>△ACC'</note>, <note>side AC and side AC'</note> are equal. Therefore, according to the properties of the isosceles triangle, the angles opposite the equal sides are equal, that is, <note>∠ACC' = ∠AC'C</note>.

Triangle Angle Sum Theorem

<note>角ACC</note>, <note>角CC'A</note>, and <note>角C'AC</note> are the three interior angles of triangle <note>ACC'</note>. According to the Triangle Angle Sum Theorem, <note>角ACC' + 角CC'A + 角C'AC = 180°</note>.

Exterior Angle Theorem of a Triangle

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

In <note>triangle B'C'C</note>, <note>angle B'C'C and angle B'CC'</note> are the two interior angles that are not adjacent to <note>angle C'B'A</note>. According to the Exterior Angle Theorem of a Triangle, <note>∠C'B'A = ∠B'C'C + ∠B'CC'</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 text: <note>The interior angle of polygon B'C'C is ∠CB'C'</note>, extending the adjacent sides of this interior angle <note>CB'</note> and <note>B'C'</note> forms the <note>angle ∠AB'C'</note> which is called <note>the exterior angle of the interior angle ∠CB'C'</note>.


Question: As shown in the figure, in Rt△ABC, ∠B=90°, AB=6, AC=10, △ABC is folded along ED, making point C coincide with point A. Then the perimeter of △ABE equals ()

Answer: 14

1. Given in the right triangle ABC, ∠B=90°, AB=6, AC=10, use the Pythagorean theorem to calculate the length of BC.

2. According to the Pythagorean theorem, ##AC^2 = AB^2 + BC^2##, substitute the data to get ##10^2= 6^2 + BC^2##, solve to get BC = √(##10^2 - 6^2##) = √(100 - 36) = √64 = 8.

3. Since △ABC is folded along ED, making point C coincide with point A, AE=EC.

4. Since AE=EC, the perimeter of △ABE can be expressed as AB+BE+##EC##.

####

##5##. Therefore, the perimeter of △ABE is AB+BE+EC=AB+BC.

##6##. Substitute the known AB and the calculated BC, the perimeter of △ABE is 6+8=14 (cm).

##7##. Through the above reasoning, the final answer is 14 (cm).

直角三角形; 垂直平分线; 对称; 垂线; 中点

Definition of Right Triangle

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

Pythagorean Theorem

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

Perimeter Theorem of Triangle

The perimeter of a triangle is equal to the sum of the lengths of its three sides. That is, if the lengths of the three sides of the triangle are denoted as a, b, and c respectively, then the perimeter P is given by P = a + b + c.

The original text: <note>The three sides of triangle △ABE are AB, BE, and AE</note>, according to the Perimeter Theorem of Triangle, the perimeter of the triangle is equal to the sum of the lengths of its three sides, that is, <note>Perimeter P = AB + BE + AE</note>.


Question: As shown in the figure, in △ABC, DE is the perpendicular bisector of AC, AE = 3cm, the perimeter of △ABD is 13cm, then the perimeter of △ABC is ()

Answer: 19cm

1. Given that DE is the perpendicular bisector of AC, and AE = 3 cm.

2. According to the properties of the perpendicular bisector, AC is bisected, therefore AE = CE = 3 cm, AD = DC.

3. Given that the perimeter of △ABD is 13 cm, ##according to the triangle perimeter formula, we have## AB + BD + AD = 13 cm.

4. ##According to the triangle perimeter formula, we have## the perimeter of △ABC = AB + BC + AC = AB + BD + DC + 2 * AE.

5. Since AD = DC, replacing AD with DC, in △ABC we have the perimeter = AB + BD + DC + 2 * 3 cm.

6. From the above, the perimeter of △ABC = AB + BD + AD + 6 cm = 13 cm + 6 cm.

7. Through the above reasoning, the final answer is 19 cm.

垂直平分线; 等腰三角形; 线段; 普通三角形

Definition of Perpendicular Bisector

In the figure of this problem, DE passes through the <note>midpoint E</note> of segment AC, and <note>DE is perpendicular to segment AC</note>. Therefore, DE is the perpendicular bisector of segment AC. Specifically, <note>AE = CE = 3 cm</note>.

In the figure of this problem, <note>the perpendicular bisector of segment AC is line DE</note>, <note>point D is on line DE</note>. According to the properties of the perpendicular bisector, <note>the distance from point D to the endpoints A and C of segment AC is equal</note>, that is, <note>AD = DC</note>,####<note>AE = CE</note>.##

Formula for the Perimeter of a Triangle

<note>The three sides of triangle ABC are AB, AC, and BC</note>. According to the formula for the perimeter of a triangle, the perimeter of the triangle is equal to the sum of the lengths of its three sides, that is, the perimeter <note>P=AB+AC+BC</note>. Similarly, <note>the three sides of triangle ABD are AB, AD, and BD</note>. According to the formula for the perimeter of a triangle, the perimeter of the triangle is equal to the sum of the lengths of its three sides, that is, the perimeter <note>P=AB+AD+BD</note>.


Question: As shown in the figure, C and D are two points on the line segment AB. If CB=4cm, DB=7cm, and D is the midpoint of AC, then AB=()

Answer: 10cm

1. Given CB=4 cm, DB=7 cm, according to the method of calculating the sum and difference of segments, we get CD=7 cm - 4 cm = 3 cm.

2. D is the midpoint of AC, according to the definition of midpoint, we know AD = CD = 3 cm.

3. According to the property of segment addition, AB = AD + DB = 3 cm + 7 cm = 10 cm.

4. Through the above reasoning, we finally obtain the length of AB as 10 cm.

Definition of Line Segment

A line segment is a portion of a straight line that includes two endpoints and all the points between them.

In the figure of this problem, <note>line segment AB is a part of a straight line, including endpoints A and B and all points between them</note>. <note>Line segment AB has two endpoints, which are A and B respectively</note>, and <note>every point on line segment AB is located between endpoints A and B</note>. <note>Line segment AC is a part of a straight line, including endpoints A and C and all points between them</note>. <note>Line segment AC has two endpoints, which are A and C respectively</note>, and <note>every point on line segment AC is located between endpoints A and C</note>. <note>Line segment AD is a part of a straight line, including endpoints A and D and all points between them</note>. <note>Line segment AD has two endpoints, which are A and D respectively</note>, and <note>every point on line segment AD is located between endpoints A and D</note>. <note>Line segment CD is a part of a straight line, including endpoints C and D and all points between them</note>. <note>Line segment CD has two endpoints, which are C and D respectively</note>.

Midpoint of a Line Segment

In the figure of this problem, the midpoint of the line segment <note>AC is point D</note>. According to the definition of the midpoint of a line segment, <note>point D divides the line segment AC into two equal parts, that is, the lengths of line segment AD and line segment DC are equal</note>. That is, <note>AD = DC</note>.


Question: In the △ABC shown in the figure, ∠ACB=90°, AD bisects ∠BAC and intersects BC at D, DE is perpendicular to AB at E. If DE=1.5cm and BD=3cm, then BC=()

Answer: 4.5cm

1. Given △ABC, ∠ACB=90°, AD bisects ∠BAC and intersects BC at D, DE is perpendicular to AB at E. According to the definition of perpendicular lines, ∠DEA=90°.

2. Since AD is the angle bisector of ∠BAC, according to the definition of angle bisector, ∠DAE=∠DAC.

####

3. In ΔADE and ΔADC, because ∠DAE=∠DAC, ∠DEA=∠DCA=90°, AD is the opposite side of ∠DEA and ∠DCA, so according to the congruent triangles theorem (AAS), ΔADE is congruent to ΔADC.

4. According to the definition of congruent triangles, DE=DC.

####

5. Since DC is equal to DE and equals 1.5cm, BD is equal to 3cm, so BC=BD+DC=3cm+1.5cm=4.5cm.

6. Through the above reasoning, the final answer is 4.5cm.

直角三角形; 垂线; 线段; 等腰三角形

Definition of Angle Bisector

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

In this problem diagram, <note>the vertex of angle BAC is point A</note>, <note>a line AD is drawn from point A</note>, <note>this line divides angle BAC into two equal angles, that is, angle BAD and angle CAD are equal</note>. Therefore, <note>line AD is the angle bisector of angle BAC</note>.

Congruence Theorem for Triangles (AAS)

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

Original: <note>三角形ADE和三角形ADC中</note>，<note>角DEA等于角DCA</note>，<note>角DAE等于角DAC</note>，<note>边AD等于AD</note>。由于这两个三角形的两角及一角的对边对应相等，根据<note>全等三角形判定定理的角角边准则(AAS)</note>，可以得出<note>三角形ADE全等于三角形ADC</note>。

Translation: <note>In triangles ADE and ADC</note>, <note>angle DEA is equal to angle DCA</note>, <note>angle DAE is equal to angle DAC</note>, <note>side AD is equal to AD</note>. Since the two triangles have two angles and the side opposite one of these angles equal, according to the <note>Congruence Theorem for Triangles (AAS)</note>, it can be concluded that <note>triangle ADE is congruent to triangle ADC</note>.

Definition of Perpendicular Lines

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

<note>Line DE and line AB intersect to form an angle ∠DEA of 90 degrees</note>, therefore according to the definition of perpendicular lines, <note>line DE and line AB are perpendicular to each other</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 ADE and triangle ADC are congruent triangles</note>, <note>side AE = side AC</note> <note>side AD = side AD</note> <note>side DE = side DC</note>, meanwhile, <note>angle DEA = angle DCA</note> <note>angle DAE = angle DAC</note> <note>angle ADE = angle ADC</note>.


Question: As shown in the figure, segment AB=20, C is the midpoint of AB, D is a point on CB, E is the midpoint of DB, and EB=3, then CD equals ()

Answer: 4

1. Given that E is the midpoint of DB, and EB=3, according to the definition of midpoint, BD=2×EB=2×3=6.

2. Given that segment AB=20, C is the midpoint of AB, according to the definition of midpoint, CB=AC=AB/2=20/2=10.

3. From CB=10 and BD=6, we get CD=CB-BD=10-6=4.

Midpoint of a Line Segment

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

In the figure of this problem, <note>the midpoint of line segment AB is point C</note>. According to the definition of the midpoint of a line segment, <note>point C divides line segment AB into two equal parts</note>, that is, <note>the lengths of line segments AC and CB are equal</note>. That is, AC = CB = 10. <note>The midpoint of line segment DB is point E</note>, according to the definition of the midpoint of a line segment, <note>point E divides line segment DB into two equal parts</note>, that is, <note>the lengths of line segments DE and EB are equal</note>. That is, DE = EB = 3.

Definition of Line Segment

<note>Line segment AB is a part of a straight line</note>, including endpoint <note>A</note> and endpoint <note>B</note> and all points in between. Line segment <note>AB</note> has two endpoints, which are <note>A</note> and <note>B</note>, and every point on line segment <note>AB</note> lies between endpoint <note>A</note> and endpoint <note>B</note>.


Question: As shown in the figure, the diagonals AC and BD of rectangle ABCD intersect at point O. CE∥BD, DE∥AC. If AB=4 and BC=3, then the perimeter of quadrilateral CODE is ()

Answer: 10

1. Given CE∥BD and DE∥AC, according to ##the definition of parallelogram##, it can be concluded that quadrilateral CODE is a parallelogram.

2. Since quadrilateral ABCD is a rectangle, according to ##the properties of the diagonals of a rectangle##, it can be deduced that ##AC=BD,## OC=OD.

3. Calculate the lengths of the diagonals AC and BD of rectangle ABCD. ##According to the definition of a rectangle, it is known that ∠ABC=90°, and based on the definition of a right triangle, it can be concluded that triangle ABC is a right triangle. Given that## AB=4 and BC=3, according to the Pythagorean theorem, AC=BD=##√(AB^2+BC^2)=√(4^2+3^2)##=5.

4. Since AC and BD intersect at point O, and OC=OD, ##according to the properties of the diagonals of a rectangle,## it can be deduced that OC=OD=(1/2)AC=2.5.

5. Since quadrilateral CODE is a parallelogram and OC=OD, ##according to the definition of a rhombus,## it can be concluded that quadrilateral CODE is a rhombus.

6. The side length of rhombus CODE is equal to OC or OD, which is 2.5. ##According to the formula for the perimeter of a rhombus,## the perimeter of the rhombus is equal to 4 times the side length.

7. The perimeter of quadrilateral CODE is 2.5×4=10.

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

矩形; 对称; 平行线; 线段; 平行四边形

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 (≅).

In the figure of this problem, due to <note>CE∥BD</note> and <note>DE∥AC</note>, the two pairs of opposite sides of quadrilateral CODE are parallel. Specifically, side CE is parallel and equal to <note>side DO</note>, and side DE is parallel and equal to <note>side OC</note>. Therefore, quadrilateral CODE is a parallelogram.

Definition of Rhombus

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

In the quadrilateral <note>四边形CODE</note>, all sides <note>CO、OD、DE、EC</note> are equal, therefore <note>四边形CODE是一个菱形</note>.

Property of Diagonals in a Rectangle

In a rectangle, the diagonals are equal in length and bisect each other.

In the figure of this problem, in the rectangle <note>ABCD</note>, side <note>AB</note> is parallel and equal to side <note>CD</note>, and side <note>AD</note> is parallel and equal to side <note>BC</note>. The diagonals <note>AC</note> and <note>BD</note> are equal and bisect each other, that is, the intersection point <note>O</note> of the diagonals <note>AC</note> and <note>BD</note> is the midpoint of both diagonals. Therefore, segment <note>OC</note> is equal to segment <note>OD</note>, and segment <note>OA</note> is equal to segment <note>OB</note>.

Pythagorean Theorem

<note>In right triangle ABC</note>, ∠ABC is a right angle (90 degrees), sides AB and BC are the legs, side AC is the hypotenuse, so according to the Pythagorean Theorem, <note>AC^2 = AB^2 + BC^2, that is, AC = √(AB^2 + BC^2) = √(4^2 + 3^2) = 5</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 ABCD is a rectangle</note>, with interior angles <note>∠ABC, ∠BCA, ∠CDA, ∠DAB all being right angles (90 degrees)</note>, and sides <note>AB and DC are parallel and equal in length</note>, sides <note>DA and CB are parallel and equal in length</note>.

Definition of Right Triangle

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

Perimeter Formula for Rhombus

The perimeter of a rhombus is equal to four times the length of one of its sides, i.e., \(C = 4 \cdot a\).

In the context of <note>Rhombus CODE</note>, <note>side OC is the side of the rhombus</note>, according to the perimeter formula for a rhombus, the perimeter of the rhombus is equal to four times its side length, that is, C = 4x<note>OC</note>.


Question: As shown in the figure, given ∠1=40°, ∠A+∠B=140°, then the degree of ∠C+∠D is ()

Answer: 80°

1. Given that the measure of angle 1 is 40°, ##connect CD, let angle CDE be angle 2, angle DCE be angle 3,## according to the triangle angle sum theorem, the sum of the interior angles of a triangle is equal to 180°, thus we can conclude angle 1 + angle 2 + angle 3 = 180°.

2. According to the triangle angle sum theorem, substitute the measure of angle 1, we get angle 2 + angle 3 = 180° - 40° = 140°.

3. ##According to the quadrilateral angle sum theorem, it is known that the sum of the interior angles of a quadrilateral is equal to 360°, thus## the sum of the interior angles of quadrilateral ABCD## is 360°, that is angle A + angle B + angle ADC + angle BCD = 360°.

4. Given that the measure of angle A + angle B is 140°, it follows that angle ADC + angle BCD = 360° - 140° = 220°.

5. The sum of angles BCE and ADE within quadrilateral ABCD## should be 220° minus the sum of angles 2 and 3 already calculated, that is angle BCE + angle ADE = 220° - 140° = 80°.

6. Through the above reasoning, it is finally concluded that the sum of the measures of angle C and angle D is 80°.

普通三角形; 三角形的外角; 邻补角

Triangle Angle Sum Theorem

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

In <note>triangle CDE</note>, <note>angle CED</note>, <note>angle CDE</note>, and <note>angle DCE</note> are the three interior angles of <note>triangle CDE</note>. According to the Triangle Angle Sum Theorem, <note>angle CED</note> + <note>angle CDE</note> + <note>angle DCE</note> = 180°. Specifically, we know <note>angle CED</note> = 40°, so we can conclude: <note>angle CDE</note> + <note>angle DCE</note> = 140°.

In any quadrilateral, the sum of the four interior angles is 360°.

In <note>quadrilateral ABCD</note>, <note>angle ADC, angle DCB, angle CBA, and angle BAD</note> are the four interior angles of the quadrilateral. According to the Sum of Interior Angles of a Quadrilateral Theorem, the sum of these four interior angles is 360°, that is, <note>angle ADC + angle DCB + angle CBA + angle BAD = 360°</note>.

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 CDE</note> is a geometric figure composed of three <note>non-collinear points C, D, E</note> and their <note>connecting line segments CD, CE, DE</note>. <note>Points C, D, E</note> are the <note>three vertices</note> of the triangle, and <note>line segments CD, CE, DE</note> are the triangle's <note>three sides</note>.

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</note> is composed of four line segments <note>AB, BC, CD, and DA</note>, which are called the sides of the quadrilateral. Quadrilateral ABCD has four vertices, namely <note>point A, point B, point C, and point D</note>, and it has four interior angles, namely <note>angle ABC, angle BCD, angle CDA, and angle DAB</note>.


Question: As shown in the figure, in the right triangle △ABC, ∠C=90°, ∠A=30°, BC=2, the radius of ⊙C is 1, point P is a point on the hypotenuse AB, through point P, a tangent PQ to ⊙C is drawn (point Q is the point of tangency), then the minimum value of the segment PQ is ()

Answer: √{2}

1. Given that point P is a point on the hypotenuse AB, a tangent PQ to ⊙C is drawn through point P (point Q is the point of tangency), connect CP and CQ.

2. ∵ PQ is a tangent to ⊙C, ##according to the property of the tangent to a circle, the tangent is perpendicular to the radius at the point of tangency## ∴ CQ ⊥ PQ, i.e., ∠CQP = 90°.

####

##3. According to the distance from a point to a line, when CP ⊥ AB, CP is minimized. Simultaneously, according to the Pythagorean theorem, PQ^2 = CP^2 - CQ^2. Given that CQ is a fixed value, equal to the radius of the circle, therefore when CP is minimized, the segment PQ is shortest.##

##4##. ####In the right triangle ACB, ∠A = 30°, ∠C = 90°, BC = 2,

##5##. Using the properties of a 30°-60°-90° triangle, we get AB = 2BC = 4, AC = BC * √3 = 2√3.

##6##. ##According to the area formula of a right triangle: S△ABC = 1/2 * AC * BC = 1/2 * 2√3 * 2 = 2√3. Using the triangle's area formula with CP and AB sides to find the area of △ABC,## when CP ⊥ AB, CP is a height of △ABC, ##i.e., S△ABC = 1/2 * AB * CP = 1/2 * 4 * CP = 2CP. Simplified as S△ABC = 2CP, and since S△ABC = 2√3, then 2√3 = 2CP, finally we get CP = √3.##

##7##. ##Given that## the radius of ⊙C, CQ = 1.

##8##. ##According to the Pythagorean theorem, CQ^2 + PQ^2 = CP^2, i.e., PQ = √(CP^2 - CQ^2) = √(3 - 1) = √2.##

##9##. Through the above reasoning, the final answer is √2.

直角三角形; 切线; 圆; 线段; 三角形的外角

Definition of Right Triangle

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

<note>In the figure of this problem</note>, in triangle ABC, <note>angle ACB is a right angle (90 degrees)</note>, so triangle ABC is a right triangle. <note>Side AC and side BC are the legs, and side AB is the hypotenuse</note>. <note>PQ is the tangent to circle C</note>, so <note>PQ is perpendicular to the radius CQ of circle C, that is, ∠CQP=90°</note>. In triangle CQP, <note>angle CQP is a right angle (90 degrees)</note>, so triangle CQP is a right triangle. <note>Side CQ and side PQ are the legs, and side CP is the hypotenuse</note>.

Definition of Tangent to a Circle

A line is a tangent to a circle if and only if it has exactly one point of intersection with the circle. This point of intersection is called the point of tangency.

The circle <note>⊙C</note> and the line <note>PQ</note> have exactly one common point <note>Q</note>, which is called the <note>tangent point Q</note>. Therefore, the line <note>PQ</note> is the <note>tangent line of circle ⊙C</note>.

Property of the Tangent Line to a Circle

A tangent line to a circle is perpendicular to the radius that passes through the point of tangency.

<note>In circle C, point Q is the point of tangency of line PQ with the circle</note>, <note>segment CQ is the radius of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent line PQ is perpendicular to the radius CQ at the point of tangency Q</note>, that is, <note>∠CQP=90°</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>In the figure of this problem, in the 30°-60°-90° triangle ABC, angle CAB is 30 degrees</note>, <note>angle CBA is 60 degrees</note>, <note>angle ACB is 90 degrees</note>. <note>Side AB is the hypotenuse</note>, <note>side BC is the side opposite the 30-degree angle</note>, <note>side AC is the side opposite the 60-degree angle</note>. According to the properties of the 30°-60°-90° triangle, <note>side BC is half of side AB</note>, <note>side AC is √3 times side BC</note>. That is: <note>BC = 1/2 * AB</note>, <note>AC = BC * √3</note>.

Area Formula of a Triangle

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

In <note>triangle ABC</note>, <note>side AB is the base</note>, <note>segment CP is the height</note>. According to the area formula of a triangle, the area of triangle ABC is equal to <note>base AB</note> multiplied by <note>height CP</note> and then divided by 2, that is, <note>Area = (AB * CP) / 2</note>.

Area of Right Triangle

The area of a right triangle is equal to half the product of the two legs that form the right angle, i.e., Area = 1/2 * base * height.

In the figure of this problem, in <note>right triangle ABC</note>, <note>angle ACB is a right angle (90 degrees)</note>, <note>sides BC and AC are the legs</note>, one leg serves as the base, and the other leg serves as the height, so the <note>area of the right triangle</note> is equal to half the product of these two legs, that is, area = 1/2 * <note>side BC</note> * <note>side AC</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 CQP, angle CQP is a right angle (90 degrees)</note>, <note>sides CQ and PQ are the legs</note>, <note>side CP is the hypotenuse</note>, so according to the Pythagorean Theorem, CP^2 = CQ^2 + PQ^2.

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 C, <note>point C is the center of the circle</note>, <note>point Q is any point on the circle</note>, <note>line segment CQ is the line segment from the center to any point on the circle</note>, therefore <note>line segment CQ is the radius of the circle</note>.

Definition of Altitude

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

In the figure of this problem, the line segment <note>CP</note> perpendicular to the <note>opposite side AB (or its extension)</note> from <note>vertex C</note> is the <note>altitude from vertex C</note>. The <note>line segment CP forms a right angle (90 degrees) with side AB (or its extension)</note>, indicating that <note>line segment CP is the perpendicular distance from vertex C to the opposite side AB (or its extension)</note>.

Distance from a Point to a Line

The distance from a point to a line is defined as the shortest distance from a point not on the line to the line, which is the perpendicular distance.

In the figure of this problem, <note>point C is a point outside line AB</note>, construct <note>the perpendicular segment CP from point C to line AB</note>. According to the definition of the distance from a point to a line, <note>the length of the perpendicular segment CP is the shortest distance from point C to line AB, i.e., d=CP</note>.


Question: As shown in the figure, it is known that line a∥b∥c, line d is perpendicular to them and intersects at points A, B, and C. If AB=3 and AC=8, then the distance between parallel lines b and c is ()

Answer: 5

1. Given parallel lines a∥b∥c, and line d is perpendicular to them and intersects at points A, B, and C.

2. According to the ##vertical distance between parallel lines##, the length of segment AB is 3, and AB is the distance between parallel lines a and b.

3. ##According to the vertical distance between parallel lines##, the length of segment AC is 8, and AC is the distance between parallel lines a and c.

4. ##According to the definition of a segment##, the length of segment BC can be obtained by subtracting the length of segment AB from the length of segment AC, i.e., BC = AC - AB.

5. Substitute the known data to get BC = 8 - 3.

6. Calculate to get the length of BC as 5, which means the distance between line b and line c is 5.

平行线; 垂线; 直线; 线段

Definition of Parallel Lines

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

In the figure of this problem, <note>lines a, b, and c lie in the same plane</note>, <note>and they do not intersect</note>, therefore, according to the definition of parallel lines, <note>lines a, b, and c are parallel, i.e., a ∥ b ∥ c</note>.

Definition of Line Segment

A line segment is a portion of a straight line that includes two endpoints and all the points between them.

<note>Line segment AB is a part of a straight line, including endpoint A and endpoint B and all points in between</note>.<note>Line segment AC is a part of a straight line, including endpoint A and endpoint C and all points in between</note>.<note>Line segment BC is a part of a straight line, including endpoint B and endpoint C and all points in between</note>.

Perpendicular Distance Between Parallel Lines

The perpendicular distance between parallel lines is defined as the length of the perpendicular segment from any point on one line to the other line.

<note>AB is the distance between parallel lines a and b</note>, <note>AC is the distance between parallel lines a and c</note>. <note>BC is the distance between parallel lines b and c</note>.


Question: As shown in the figure, AB is the diameter of ⊙O, O is the center of the circle, chord CD ⊥ AB at E, AB = 10, CD = 8, then the length of OE is ()

Answer: 3

1. Connect OC. Given that AB is the diameter of ⊙O, AB = 10, the distance from the center O to any point on the circumference is the radius, so OC = radius = 10 / 2 = 5.

2. Given that the chord CD is perpendicular to the diameter AB and intersects at point E, since CD ⊥ AB, then ∠OEC = 90°. According to the perpendicular bisector theorem, CD is perpendicularly bisected by the diameter AB, i.e., CE = ED = CD / 2 = 8 / 2 = 4.

3. Triangle OEC is a right triangle, where ∠OEC = 90°, OC = 5, CE = 4.

4. According to the Pythagorean theorem (in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs). In triangle OEC, OE² + CE² = OC².

5. Substitute the known values, OE² + 4² = 5².

6. Solve this equation, OE² + 16 = 25.

7. Thus, OE² = 25 - 16 = 9.

8. So OE = √9 = 3.

9. Therefore, the length of OE is 3.

圆; 弦; 垂线; 直角三角形; 线段

Definition of Diameter

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

<note>AB is the diameter</note>, connecting the <note>center O</note> and <note>points A and B on the circumference</note>, with a length of <note>2 times the radius</note>, that is, <note>AB = 2 * radius = 10</note>.

Definition of Radius

<note>Point O is the center of the circle</note>, <note>Point C is any point on the circle</note>, <note>Line segment OC is the line segment from the center to any point on the circle</note>, therefore <note>Line segment OC is the radius of the circle</note>.

Definition of Chord

A chord is defined as a line segment that connects any two points on a circle.

<note>In circle O, points C and D are any two points on the circle</note>, <note>line segment CD connects these two points</note>, so <note>line segment CD is a chord of circle O</note>.

Definition of Right Triangle

<note>In triangle OEC</note>, <note>angle OEC is a right angle (90 degrees)</note>, therefore <note>triangle OEC is a right triangle</note>. <note>Side OE and side CE are the legs</note>, <note>side OC is the hypotenuse</note>.

Perpendicular Diameter Theorem

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

<note>In the diagram of this problem</note>, in circle O, <note>diameter AB is perpendicular to chord CD</note>, then according to the Perpendicular Diameter Theorem, <note>diameter AB bisects chord CD</note>, that is, <note>CE = ED = CD / 2 = 4</note>, and <note>diameter AB bisects the two arcs subtended by chord CD</note>, that is, <note>arc AC = arc AD</note>.

Pythagorean Theorem

In the <note>right triangle OEC</note>, <note>angle OEC is a right angle (90 degrees)</note>, <note>sides OE and CE are the legs</note>, <note>side OC is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>OC² = OE² + CE²</note>.


Question: As shown in the figure, C and D are points on line segment AB. If CB = 4cm, DB = 7cm, and D is the midpoint of AC, then the length of AB is equal to ()

Answer: 10cm

1. Given that the length of CB is 4 cm and the length of DB is 7 cm, according to segment subtraction, the length of CD is 3 cm.

2. Since D is the midpoint of AC, according to ##segment midpoint##, AD is equal to CD, and the length of AD is 3 cm.

3. According to segment addition, AB can be expressed as the sum of AD and DB, so the length of AB is equal to 3 cm plus 7 cm, which is 10 cm.

Definition of Line Segment

In the figure of this problem, <note>line segment AB is a part of a straight line, containing endpoints A and B and all points between them</note>. <note>Line segment CB and line segment DB are also line segments</note>, consisting of <note>all points from endpoint C to endpoint B and from endpoint D to endpoint B</note> respectively.

Midpoint of a Line Segment

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

<note>The midpoint of line segment AC is point D</note>. According to the definition of the midpoint of a line segment, <note>point D divides line segment AC into two equal parts, which means the lengths of line segments AD and DC are equal</note>. That is, <note>AD = DC</note>.