Question: As shown in the figure, in △ABC, AB=AC, D is the midpoint of BC, ∠B=40°, then ∠BAD=()

Answer: 50°

1. Given AB=AC, ##so according to the definition of an isosceles triangle, we have: triangle ABC is an isosceles triangle. Since point D is the midpoint of BC, AD is the median of triangle ABC, i.e., BD=CD.##.

2. ##According to the theorem that the altitude, median, and angle bisector of an isosceles triangle coincide:## AD is the angle bisector of the vertex angle ##BAC## of △ABC, and AD is also the perpendicular bisector of BC, so AD is perpendicular to BC.

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

4. Since AD is perpendicular to BC, i.e., ∠ADB is a right angle, ∠ADB=90°.

5. In the right triangle △ADB, according to the ##triangle angle sum theorem##, and given that ##∠ABD##=∠B=40°.

6. Therefore, ∠BAD=##180°-∠ADB-∠B=180°-90°-40°=50°##.

等腰三角形; 中点; 线段

Definition of Isosceles Triangle

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

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.

The line <note>AD</note> passes through the <note>midpoint D of segment BC</note> and <note>is perpendicular to segment BC</note>. Therefore, <note>line AD is the perpendicular bisector of segment BC</note>.

Midpoint of a Line Segment

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

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

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.

<note>In triangle ABC, vertex A is one of the vertices of the triangle</note>, <note>the opposite side BC is the side opposite vertex A</note>. <note>Point D is the midpoint of side BC</note>, <note>line segment AD is the segment from vertex A to midpoint D of the opposite side BC</note>, therefore <note>AD is a median of triangle ABC</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>The angle ∠ADB formed by the intersection of line AD and line BC is 90 degrees</note>, therefore, according to the definition of perpendicular lines, <note>line AD and line BC are perpendicular to each other</note>.

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

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

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

Triangle Angle Sum Theorem

In the figure of this problem, in <note>triangle ABD</note>, <note>angle ABD</note>, <note>angle ADB</note>, and <note>angle BAD</note> are <note>the three interior angles of triangle ABD</note>. According to the <note>Triangle Angle Sum Theorem</note>, <note>angle ABD + angle ADB + angle BAD = 180°</note>.


Question: As shown in the figure, AB is the diameter of ⊙O, ∠AOC=140°, then ∠D is ()

Answer: 20°

1. Given ∠AOC=140°, ##According to the definition of supplementary angles, ∠AOC+∠BOC=180°, then ∠BOC=180°-140°=40°.##

####

##2. According to the inscribed angle theorem, ∠BDC is an inscribed angle of the circle, equal to half of the central angle it subtends.##

##3. ##Therefore, ∠BDC=1/2 * ∠BOC = 1/2 * 40° = 20°.

##4. ##Through the above reasoning, the final answer is 20°.

Definition of Diameter

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

In the figure of this problem, <note>AB is the diameter</note>, connecting <note>the center O</note> and <note>points A and B on the circumference</note>, with a length of <note>2 times the radius</note>, i.e., <note>AB = 2 * r</note>.

Definition of Central Angle

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

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

Definition of Inscribed Angle

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

In the figure of this problem, in circle O, <note>the vertex D of angle BDC is on the circumference</note>, <note>the two sides of angle BDC intersect circle O at points B and C respectively</note>. Therefore, <note>angle BDC is an inscribed angle</note>.

Inscribed Angle Theorem

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

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

Definition of Linear Pair of Angles

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

In the figure of this problem, <note>angle AOC and angle BOC have a common side OC</note>, <note>their other sides OA and OB are extensions in opposite directions</note>, so angle AOC and angle BOC are adjacent supplementary angles.


Question: As shown in the figure, in the quadrilateral ABCD, ∠BAD=120°, ∠B=∠D=90°, if points M and N are found on BC and CD respectively, such that the perimeter of △AMN is minimized, then the degree of ∠AMN+∠ANM is ()

Answer: 120°

1. As shown in the figure, find points M on BC and N on CD such that the perimeter of △AMN is minimized. Construct the symmetric points of A with respect to BC and CD, denoted as A' and A''. Connect A' and A'', intersecting BC at M and CD at N. Since the shortest distance between two points is a straight line, A'A'' represents the minimum perimeter of △AMN.

2. Given ∠B = ∠D = 90°, and A' and A'' are the symmetric points of A with respect to BC and CD, respectively. AB and AD are the distances to BC and CD, respectively. According to the properties of symmetry, points A, A', and B are collinear; points A, A'', and C are collinear. Therefore, ∠BAD = ∠A'AA'' = 120°.

3. In △A'AA'', ∠A'AA'' is 120°. According to the triangle angle sum theorem, ∠AA'A'' + ∠AA''A' = 180° - ∠A'AA'' = 180° - 120° = 60°.

4. Since A' is the symmetric point of A with respect to BC, according to the properties of symmetry, AM = A'M. Therefore, according to the properties of an isosceles triangle, ∠BA'M = ∠BAM.

5. Similarly, A'' is the symmetric point of A with respect to CD. According to the properties of symmetry, AN = A''N. Therefore, according to the properties of an isosceles triangle, ∠NAD = ∠NA''D.

6. Given ∠BAM + ∠NAD = ∠NA''D + ∠BA'M = ∠AA'A'' + ∠AA''A' = 60°.

7. Therefore, according to the triangle angle sum theorem, ∠AMN + ∠ANM = 180° - ∠MAN; and ∠MAN = ∠BAD - ∠BAM + ∠NAD = 120° - 60° = 60°.

8. After the above reasoning, it is finally concluded that ∠AMN + ∠ANM = 120°.

垂直平分线; 反射; 直角三角形; 邻补角; 普通三角形

Triangle Angle Sum Theorem

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

In the figure of this problem, <note>in triangle AA'A'', ∠AA'A'', ∠AA''A, and ∠A'AA'' are the three interior angles of triangle AA'A''</note>, according to the Triangle Angle Sum Theorem, <note>∠AA'A'' + ∠AA''A + ∠A'AA'' = 180°</note>, in <note>triangle AMN, ∠AMN, ∠MAN, and ∠ANM are the three interior angles of triangle AMN</note>, according to the Triangle Angle Sum Theorem, <note>∠AMN + ∠MAN + ∠ANM = 180°</note>.

Properties of Symmetry

In a symmetric figure, the corresponding points on either side of the axis of symmetry have equal perpendicular distances from the axis of symmetry.

In the figure of this problem, the axis of symmetry for <note>A and A’</note> is <note>BC</note>, and the axis of symmetry for <note>A and A‘’</note> is <note>DC</note>. Because the distance from any point on the axis of symmetry to the symmetric point is equal, <note>AM=A‘M</note>, <note>AN=A’‘N</note>, <note>AB=BA‘</note>, <note>AD=DA’‘</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.

In the figure of this problem, <note>in the isosceles triangle AA’M, sides AM and A'M are equal.</note> Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, that is, <note>∠BA'M = ∠BAM</note>. In <note>the isosceles triangle AA’’N, sides AN and A''N are equal.</note> Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, that is, <note>∠NAD = ∠NA''D</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 AA'M, side AM and side A'M are equal</note>, therefore triangle AA'M is an isosceles triangle, <note>in triangle AA''N, side AN and side A'N are equal</note>, therefore triangle AA'N is an isosceles triangle.


Question: As shown in the figure, the parallelogram ABCD is divided into 4 parallelograms. It is known that the areas of three of them are 8, 10, and 30 respectively. Then the area of the fourth parallelogram is ()

Answer: 24

1. Given that parallelogram ABCD is divided into 4 parallelograms, ##let EF intersect AD and BC at points E and F; MN intersect AB and CD at points M and N, and EF intersect MN at point O,## according to the problem statement AB∥EF∥CD, AD∥MN∥BC.

2. ##Parallelogram AEOM## and parallelogram MBFO have the same height, with bases OE and OF respectively.

3. Parallelogram EOND and parallelogram OFCN have the same height, with bases OE and OF respectively.

4. According to the area calculation formula of parallelograms, ##for parallelograms with the same height,## the ratio of areas is equal to the ratio of bases, i.e., ##rac{S_{	ext{parallelogram AEOM}}}{S_{	ext{parallelogram MBFO}}} = rac{OE}{OF}##.

5. Given ##S_{	ext{parallelogram AEOM}} = 8## and ##S_{	ext{parallelogram MBFO}} = 10##, so ##rac{8}{10} = rac{4}{5}##, i.e., ##rac{OE}{OF} = rac{4}{5}##.

6. Therefore, ##rac{S_{	ext{parallelogram EOND}}}{S_{	ext{parallelogram OFCN}}} = rac{OE}{OF} = rac{4}{5}##.

7. Given ##S_{	ext{parallelogram OFCN}} = 30##, then ##rac{S_{	ext{parallelogram EOND}}}{30} = rac{4}{5}##.

8. Through calculation, ##S_{	ext{parallelogram EOND}} = 30 	imes rac{4}{5} = 24##.

9. Through the above reasoning, the final answer is 24.

Definition of Parallelogram

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

<note>Quadrilateral ABCD</note>, <note>Quadrilateral AEOM</note>, <note>Quadrilateral MBFO</note>, and <note>Quadrilateral EOND</note> are parallelograms. In quadrilateral ABCD, 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>. In <note>Quadrilateral AEOM</note>, side <note>AE</note> is parallel and equal to side <note>MO</note>, and side <note>AM</note> is parallel and equal to side <note>EO</note>. In quadrilateral MBFO, side <note>MB</note> is parallel and equal to side <note>FO</note>, and side <note>MO</note> is parallel and equal to side <note>BF</note>. In quadrilateral EOND, side <note>EO</note> is parallel and equal to side <note>ND</note>, and side <note>ED</note> is parallel and equal to side <note>ON</note>.

Area Formula of a Parallelogram

The area of a parallelogram is calculated by multiplying the length of its base by the corresponding height, which can be expressed as \( A = \text{base length} \times \text{height} \).

<note>Parallelogram AEOM</note> and parallelogram <note>MBFO</note> have the same height, with their bases being <note>OE</note> and <note>OF</note> respectively. Therefore, the ratio of their areas is equal to the ratio of their bases, that is, <note>\( \frac{S_{\text{parallelogram AEOM}}}{S_{\text{parallelogram MBFO}}} = \frac{OE}{OF} \)</note>. Specifically, <note>\( \frac{8}{10} = \frac{4}{5} \)</note>. Similarly, parallelogram <note>EOND</note> and parallelogram <note>OFCN</note> have the same height, with their bases being <note>OE</note> and <note>OF</note> respectively. Therefore, the ratio of their areas is equal to the ratio of their bases, that is <note>\( \frac{S_{\text{parallelogram EOND}}}{S_{\text{parallelogram OFCN}}} = \frac{OE}{OF} \)</note>. Specifically, <note>\( \frac{S_{\text{parallelogram EOND}}}{30} = \frac{4}{5} \)</note>.


Question: Given: As shown in the figure, AB∥CD, BC bisects ∠ABD, and ∠C=40°, then the degree of ∠D is ()

Answer: 100°

1. Given: AB∥CD and ∠C=40°.

2. According to ##Parallel Lines Axiom 2##, we get: ∠ABC=∠C=40°.

3. Since BC bisects ∠ABD, according to the definition of angle bisector, we get: ∠DBC=∠ABC=40°.

4. In △BCD, according to the triangle angle sum theorem, we get: ∠D=180°-∠DBC-∠C.

5. Substituting the given conditions, we calculate: ∠D=180°-40°-40°=100°.

6. Through the above reasoning, the final answer is ∠D=100°.

平行线; 内错角; 同旁内角; 三角形的外角

Definition of Angle Bisector

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

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

Parallel Postulate 2 of Parallel Lines

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

<note>In the diagram of this problem</note>, two parallel lines <note>AB and CD</note> are intersected by a third <note>line BC</note>, forming the following geometric relationships: 1. <note>Alternate interior angles</note>: <note>∠ABC and ∠BCD are equal</note>. <note>∠ABC = ∠C = 40°</note>.

Triangle Angle Sum Theorem

In the diagram of this problem, the interior angles of triangle BCD are ∠DBC, <note>∠BDC</note>, and ∠C. According to the Triangle Angle Sum Theorem, ∠DBC + <note>∠BDC</note> + ∠C = 180°.

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


Question: As shown in the figure, in Rt△ABC, ∠BAC=90°, △ABC is rotated 90° clockwise around point A to obtain △AB′C′ (point B's corresponding point is point B′, point C's corresponding point is point C′). Connect CC′. If ∠CC′B′=32°, then the measure of ∠AC′B′ is ()

Answer: 13°

1. Given in Rt△ABC, ∠BAC=90°. According to the definition of rotation, △ABC is rotated 90° clockwise around point A to obtain △AB'C'.

2. According to the properties of rotation, AC=AC'.

3. According to the properties of rotation, ∠BAC rotated 90° clockwise to obtain ∠B'AC', so ∠B'AC'=90°.

4. In addition, ∠C'AC=90°, according to the definition of isosceles right triangle, △ACC' is an isosceles right triangle. According to the triangle sum theorem, ∠ACC'=∠AC'C=(180°-∠C'AC)/2=(180°-90°)/2=45°.

5. Also, because ∠CC'B'=32°, so ∠AC'B'=∠ACC'-∠CC'B'=45°-32°.

6. Calculated as: ∠AC'B'=13°.

7. Through the above reasoning, the final answer is that the size of ∠AC'B' is 13°.

直角三角形; 旋转; 邻补角; 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>Angle BAC is a right angle (90 degrees)</note>, therefore triangle ABC is a right triangle. <note>Side AB and side AC are the legs</note>, <note>side BC is the hypotenuse</note>.

Triangle Angle Sum Theorem

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

Definition of Isosceles Right Triangle

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

<note>Triangle ACC' is an isosceles right triangle</note>, in which <note>angle CAC' is a right angle (90 degrees)</note>, <note>side AC and side AC' are equal right-angled sides</note>.

Properties of Rotation

1. The distance between corresponding points and the center of rotation is equal.
2. The angle between the segments connecting corresponding points and the center of rotation is equal to the angle of rotation.
3. The figure before and after rotation is congruent (≅).

<note>Knowledge Point Name: Properties of Rotation</note> <note>Knowledge Point Content: 1. The distances from corresponding points to the center of rotation are equal. 2. The angle between the line segments connecting corresponding points and the center of rotation is equal to the angle of rotation. 3. The figures before and after rotation are congruent.</note>
In the figure of this problem, the figure <note>ABC</note> is rotated around the center of rotation <note>A</note> by <note>90°</note> to obtain the figure <note>AB'C'</note>. According to the properties of rotation: 1. The distances from corresponding points (such as point <note>C</note> and point <note>C'</note>) to the center of rotation <note>A</note> are equal, i.e., <note>AC=AC'</note>; 2. The angle between the line segments connecting corresponding points and the center of rotation is equal to the angle of rotation <note>90°</note>, i.e., <note>angle CA'C' = 90°</note>; 3. The figure before rotation <note>ABC</note> and the figure after rotation <note>AB'C'</note> are congruent, i.e., <note>figure ABC ≅ figure AB'C'</note>.

Definition of Rotation

In a plane, the transformation of rotating a figure about a point O by a certain angle is called rotation. Point O is referred to as the center of rotation, and the angle of rotation is called the angle of rotation. If a point P on the figure is transformed to point P' via the rotation, then these two points are called corresponding points of the rotation.

In the figure of this problem, <note>AB'C' is the figure ABC rotated 90° clockwise (or counterclockwise) around the fixed point A</note>. Among them, <note>point C and point C_ are corresponding points</note>, <note>line segment CA and CA_ are corresponding line segments</note>, <note>∠CBA and ∠C'B'A are corresponding angles</note>, <note>point A is the center of rotation</note>, <note>the degree of ∠C'AC is called the angle of rotation</note>.


Question: As shown in the figure, points A, B, C, D are on ⊙O, ∠AOC=140°, point B is the midpoint of arc AC, then the degree of ∠D is ()

Answer: 35°

1. Given: Points A, B, C, D are on ⊙O, ##according to the definition of central angle, we know that ∠AOC is a central angle,## and ∠AOC=140°.

2. Connect OB, point B is the midpoint of arc AC, therefore the lengths of arc AB and arc BC are equal.

3. According to the properties of central angles, ##it is known that the central angle ∠AOB corresponds to arc AB, and the central angle ∠BOC corresponds to arc BC, since arc AB=arc BC, ∠AOB=∠BOC, so## ∠AOB is half of ∠AOC.

4. Since ∠AOC=140°, therefore ∠AOB=##1/2##∠AOC=70°.

5. According to the inscribed angle theorem, we know that the inscribed angle is equal to half of its corresponding central angle, that is, the inscribed angle corresponding to ##arc AB## ∠ADB=##1/2##∠AOB.

6. Since ∠AOB=70°, therefore ∠ADB=∠D=##1/2##70°=35°.

7. Through the above reasoning, the final answer is 35°.

圆; 圆心角; 圆周角; 弧; 中点

Definition of Central Angle

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

<note>∠AOC, ∠AOB, and ∠BOC are central angles</note>. Their vertex is <note>O</note>, and their sides pass through points on the circumference <note>A and C</note>, <note>A and B</note>, <note>B and C</note> respectively. According to the problem, <note>∠AOC = 140°</note>, <note>∠AOB = ∠BOC = 70°</note>.

Definition of Inscribed Angle

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

<note>In the figure of this problem, in circle O</note>, <note>the vertex D of angle ADB is on the circumference</note>, and the two sides of angle ADB intersect <note>circle O at points A and B</note> respectively. Therefore, <note>angle ADB is an inscribed angle</note>.

Inscribed Angle Theorem

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

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

Central Angle Theorem

In the same circle or in congruent circles, arcs corresponding to equal central angles are equal, chords corresponding to equal central angles are equal, and the perpendicular distances from the center to these chords are also equal.

In <note>circle O</note>, two <note>central angles subtend equal arcs, i.e., arc AB = arc BC</note>; their corresponding <note>central angles are also equal, i.e., central angle ∠AOB = ∠BOC</note>.

Definition of Arc

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

<note>There are three points A, B, and C on circle O</note>, arc AB is <note>a segment of a curve connecting point A and point B</note>, arc BC is <note>a segment of a curve connecting point B and point C</note>, arc AC is <note>a segment of a curve connecting point A and point C</note>. According to the definition of an arc, arc AB is <note>a segment of a curve between points A and B on the circle</note>, arc BC is <note>a segment of a curve between points B and C on the circle</note>, arc AC is <note>a segment of a curve between points A and C on the circle</note>.


Question: After folding a rectangular piece of paper as shown in the figure, then unfolding it. If ∠1=56°, then ∠2 equals ()

Answer: 68°

1. Given that the two overlapping angles are equal due to the folding property, ∠1 is equal to the angle it overlaps with, thus this angle is also 56°, ##we denote it as ∠3=∠1=56°##.

2. According to the definition of ##same-side interior angles, ∠1+∠3## and ∠2 form same-side interior angles.

3. Based on ##Parallel Postulate 2##, the sum of two same-side interior angles is 180°, so we can conclude 2∠1+∠2=180°.

4. Since ∠1=56°, substituting into the above formula, we can solve: 2∠1+∠2=##2×56°## + ∠2=180°.

5. Therefore, ∠2=180°-##2×56°## = 68°.

6. Through the above reasoning, we find the value of ∠2 to be 68°.

对顶角; 邻补角; 反射

Definition of Consecutive Interior Angles

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

In the figure of this problem, the upper and lower edges of the rectangular paper are cut by the fold line (transversal), <note>∠1+∠3 and ∠2 are on the same side of the fold line and within the upper and lower edges of the rectangular paper</note>, so <note>∠1+∠3 and ∠2 are consecutive interior angles</note>. Consecutive interior angles <note>∠1+∠3 and ∠2 are supplementary, that is, ∠1+∠3 + ∠2 = 180 degrees</note>.

Parallel Postulate 2 of Parallel Lines

In the figure of this problem, the two parallel sides of the rectangular paper are folded to form a crease, resulting in the following geometric relationship: <note>same-side interior angles: ∠1+∠3 and ∠2 are supplementary</note>, that is, <note>2∠1 + ∠2 = 180°</note>. Given that <note>∠1 = 56°</note>, substituting into the formula, we get <note>∠2=180°-2*56°=68°</note>.


Question: As shown in the figure, O is a point on the line AB, ∠1=40°, OD bisects ∠BOC, then the degree of ∠2 is ()

Answer: 70°

1. Given ∠AOB is a straight angle, according to the definition of a straight angle, we get ∠AOB=180°.

2. Given ∠1=40°, based on the above conclusion, we get ∠COB=180°-40°=140°.

3. Given OD bisects ∠BOC, according to the definition of an angle bisector, specifically in this problem, OD bisects ∠BOC, we get ∠BOD=∠COD.

4. Because OD bisects ∠BOC, according to the definition of an angle bisector, we get ∠BOD=∠COD=1/2∠BOC.

5. Based on step 2, we get ∠BOC=140°, thus ∠BOD=∠COD=1/2×140°=70°.

6. According to the figure, we know ∠2=∠BOD, therefore ∠2=70°.

7. Through the above reasoning, we finally get the answer as 70°.

对顶角; 邻补角; 直线; 射线

Definition of Angle Bisector

The vertex of <note>angle BOC is point O</note>, a line <note>OD is drawn from point O</note>, this line divides angle BOC into two equal angles, that is, <note>angle BOD and angle COD are equal</note>. Therefore, <note>line OD is the angle bisector of angle BOC</note>.

Definition of Straight Angle

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

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


Question: As shown in the figure, fold △ABC so that point A coincides with the midpoint D of side BC, and the crease is MN. If AB=9 and BC=6, then the perimeter of △DNB is ()

Answer: 12

1. Given that point D is the midpoint of side BC, and BC=6.

2. According to the definition of the midpoint of a line segment, side BD is equal to half of BC, i.e., BD=3.

3. According to the folding property, we can obtain that DN is equal to AN.

4. According to the formula for the perimeter of a triangle, the perimeter of △DNB is equal to DN+BN+BD.

5. Since DN=AN, AN is the line segment formed after folding, we get the perimeter of △DNB = AN+BN+BD.

6. From the relationship in the figure, AN+BN=AB, AB=9, so AN+BN=9.

7. Substituting the known values into the perimeter formula of triangle △DNB, we get the perimeter of △DNB = 9+3.

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

中点; 对称; 普通三角形

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 BC is point D</note>. According to the definition of the midpoint of a line segment, point D divides line segment BC into two equal parts, that is, <note>the lengths of line segments BD and DC are equal</note>. That is, BD = DC = 3.

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 <note>triangle DBN</note> has <note>three sides DB, BN, ND</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, <note>perimeter P = DB + BN + ND</note>.


Question: As shown in the figure, lines AB and CD are intersected by BC. If AB∥CD, ∠1=45°, ∠2=35°, then ∠3=()

Answer: 80°

2. According to the theorem of ##alternate interior angles## of parallel lines, we get ##∠BCD##=∠1=45°.

4. Let the vertex angle of ∠3 be point O. In △OCD, ∠3 as an exterior angle of △OCD is equal to the sum of the two non-adjacent interior angles of the triangle.

6. According to steps 2 and 3, ∠3 = ## 45° + 35°## = 80°.

平行线; 同位角; 内错角

Definition of Parallel Lines

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

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

Exterior Angle Theorem of Triangle

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

In the diagram of this problem, in triangle OCD, <note>∠3 is an exterior angle of the triangle</note>, <note>∠C and ∠2 are the two interior angles that are not adjacent to the exterior angle ∠3</note>. According to the <note>Exterior Angle Theorem of Triangle</note>, <note>the exterior angle ∠3 is equal to the sum of the two non-adjacent interior angles ∠C and ∠2</note>, i.e., <note>∠3 = ∠C + ∠2</note>.

Parallel Postulate 2 of Parallel Lines

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

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

Definition of Alternate Interior Angles

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

<note>Two parallel lines AB and CD</note> are intersected by a <note>line BC</note>, where <note>angle 1 and angle BCD are located between the two parallel lines and on opposite sides of the intersecting line BC</note>. Therefore, <note>angle 1 and angle BCD are alternate interior angles</note>. Alternate interior angles are equal, that is, angle 1 is equal to angle <note>BCD=45°</note>.


Question: As shown in the figure, place the right-angle vertex of a right triangle on the edge of a ruler. If ∠1=30°, then ∠2 is ()

Answer: 60°

1. Given ∠1=30°, the right angle vertex of the right triangle plate is placed on one side of the ruler, ##let the right angle point of the right triangle be ∠3+∠1, according to the definition of the right triangle##, we get ∠1 + ∠3 = 90°.

2. From the conclusion of step 1 and the given condition ∠1=30°, according to the algebraic sum of angles, we get ∠3 = 90° - ∠1 = 60°.

3. According to ##the definition of corresponding angles##, we get ∠2 and ∠3 as corresponding angles.

4. ##According to Parallel Postulate 2##, corresponding angles are equal, so ∠2 = ∠3.

5. From the conclusions of step 2 and step 4, we get ∠2 = 60°.

直角三角形; 对顶角; 邻补角

Definition of Right Triangle

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

<note>One of the angles is a right angle (90 degrees ∠1 + ∠3)</note>, and the sum of the other two interior angles is also 90°.

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>Angle 2 and Angle 3 are on the same side of the transversal, on the same side of the two intersected lines</note>, therefore <note>Angle 2 and Angle 3 are corresponding angles</note>. <note>Corresponding angles are equal</note>, that is, <note>Angle 2 is equal to Angle 3</note>.

Parallel Postulate 2 of Parallel Lines

In the figure of this problem, the two sides of the ruler are parallel to each other, and the side where the triangle board is located intersects the two parallel lines, <note>∠2 and ∠3 are corresponding angles intercepted by the third line on the parallel lines</note>. Therefore, according to the <note>corresponding angles theorem</note>, <note>∠2 = ∠3</note>.


Question: As shown in the figure, in △ABC, D and E are points on sides AB and AC respectively, DE∥BC, ∠ADE=35°, ∠C=120°, then ∠A is ()

Answer: 25°

1. Given DE∥BC, according to the parallel axiom 2 of parallel lines, it follows that ∠AED=∠C (corresponding angles are equal).

2. Given ∠C=120°, it follows that ∠AED=120°.

3. Given ∠ADE=35°, in triangle ADE, according to the triangle angle sum theorem, we get ∠ADE+∠AED+∠A=180°.

4. Substitute the known angles into the equation: 35°+120°+∠A=180°.

5. Solve the equation to get ∠A=180°-35°-120°.

6. Calculate to get ∠A=25°.

7. Through the above reasoning, the final answer is 25°.

平行线; 内错角; 普通三角形; 三角形的外角

Triangle Angle Sum Theorem

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

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

Definition of Corresponding Angles

<note>Two parallel lines DE and BC</note> are intersected by a <note>line AC</note>, where <note>angle AED</note> and <note>angle C</note> are on the same side of the intersecting line AC, on the same side of the two intersected lines <note>DE and BC</note>, therefore <note>angle AED and angle C are corresponding angles</note>. Corresponding angles are equal, that is, <note>angle AED is equal to angle C</note>.

Parallel Postulate 2 of Parallel Lines

In the figure of this problem, <note>two parallel lines BC and DE</note> are intersected by <note>the third line AC</note>, forming the following geometric relationships: 1. Corresponding angles: <note>angle AED and angle C are equal</note>. 2. Alternate interior angles: none. 3. Consecutive interior angles: none. These relationships indicate that when two parallel lines are intersected by a third line, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.


Question: As shown in the figure, it is measured that BD=120m, DC=60m, EC=50m, then the width of the river AB is ()

Answer: 100m

1. Given BD=120 meters, DC=60 meters, EC=50 meters, ##according to the definition of perpendicular lines, ∠ABD##=∠ECD=90°.

 2. Since ##∠ABD## and ∠ECD are right angles, and ∠ADB and ∠EDC are vertical angles, ##according to the definition of vertical angles,## therefore ∠ADB=∠EDC, ##and according to the theorem of similar triangles (AA),## thus △ABD is similar to △ECD.

 3. According to ##the definition of similar triangles,## the following proportional relationship is obtained: the opposite side of ##∠ADB## (AB) and the corresponding side of △ECD (EC) are proportional to the opposite side of ##∠BAD## (BD) and the corresponding side of △ECD (CD), that is ##AB##/EC = BD/CD.

 4. Substitute the known values into the proportion: AB/50 = 120/60.

 5. Solve the above proportion to get AB = (120 × 50) / 60.

 6. After calculation, AB = 100 meters.

 7. Through the above reasoning, it is concluded that the width of the river AB is 100 meters.

直角三角形; 线段; 垂线

Definition of Similar Triangles

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

In the figure of this problem, triangle ABD and triangle ECD are similar triangles. According to the definition of similar triangles: <note>angle ABD = angle ECD</note>, <note>angle ADB = angle EDC</note>, <note>angle DAB = angle DEC</note>; <note>AB/EC = BD/CD = AD/ED</note>.

Definition of Vertical Angles

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

<note>Line AE</note> and <note>Line CB</note> intersect at point D, forming four angles: ∠ADB, <note>∠ADC</note>, ∠CDE, and <note>∠EDB</note>. According to the definition of vertical angles, ∠ADB and ∠EDC are vertical angles, <note>∠ADC</note> and <note>∠EDB</note> are vertical angles. Since vertical angles are equal, ∠ADB = ∠EDC, <note>∠ADC</note> = <note>∠EDB</note>.

Definition of Right Triangle

In the diagram of this problem, <note>triangle ABD</note>, <note>angle ABD</note> is a right angle (90 degrees), so <note>triangle ABD</note> is a right triangle. Side <note>AB</note> and side <note>BD</note> are the legs, and side <note>AD</note> is the hypotenuse. Similarly, in triangle ECD, angle ECD is a right angle (90 degrees), so triangle ECD is a right triangle. Side <note>EC</note> and side CD are the legs, and side ED is the hypotenuse.

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>The angle ∠ABD formed by the intersection of line AB and line BD is 90 degrees</note>, therefore, according to the definition of perpendicular lines, line AB and line BD are perpendicular to each other. Similarly, <note>The angle ∠ECD formed by the intersection of line CE and line CD is 90 degrees</note>, therefore, according to the definition of perpendicular lines, line CE and line CD are perpendicular to each other.

Similarity Theorem for Triangles (AA)

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

<note>Angle ABD is equal to angle ECD</note>, and <note>angle ADB is equal to angle EDC</note>, so <note>triangle ABD is similar to triangle ECD</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 the length of AB is equal to ()

Answer: 10cm

1. Given BD=7cm, BC=4cm, according to subtraction, we get CD=BD-BC=3cm.

2. Since the problem states that D is the midpoint of AC, according to the definition of midpoint, we get AC=2*CD=2*3cm=6cm.

3. ##Therefore##, AB=AC+CB, that is AB=6cm+4cm=10cm.

4. Through the above reasoning, we finally get the answer AB=10cm.

Midpoint of a Line Segment

<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, that is, the lengths of line segments AD and DC are equal</note>. That is, <note>AD = DC</note>.


Question: As shown in the figure, AB⊥CD at D, DE⊥DF, if ∠BDE=60°, then ∠CDF equals ()

Answer: 60°

1. Given that line AB is perpendicular to line CD at point D, according to the definition of perpendicularity, it is known that ∠BDC and ∠ADC are both 90°.

2. Given that line DE is perpendicular to line DF at point D, according to the definition of perpendicularity, it is known that ∠EDF = 90°.

3. Also, since ∠BDE = 60°, it can be deduced that ∠CDE = 90° - ∠BDE = 90° - 60° = 30°.

4. Since ∠CDF + ∠CDE = 90°, ∠CDE = 30°, it can be indirectly concluded that ∠CDF = 90° - 30° = 60°.

5. Therefore, through the above reasoning, the final answer is 60°.

垂线; 垂直平分线; 邻补角; 直线; 对顶角

Definition of Perpendicular Lines

<note>The angle ∠BDC formed by the intersection of line AB and line CD is 90 degrees</note>, so according to the definition of perpendicular lines, line AB and line CD are perpendicular to each other. Similarly, <note>The angle ∠EDF formed by the intersection of line DE and line DF is 90 degrees</note>, so according to the definition of perpendicular lines, line DE and line DF are perpendicular to each other.


Question: As shown in the figure, a big tree breaks at point B, which is 9 meters above the ground. The top of the tree, point A, falls to a point 12 meters away from the base of the tree, point C. Find the height of the tree before it broke.

Answer: 24米

1. According to the problem, point B is 9 meters above the ground, and point A is 12 meters away from the base of the tree at point C, so BC=9, AC=12.

2. In the right triangle ABC, angle BCA is a right angle. According to the Pythagorean theorem, we have AB = ##√(BC^2 + AC^2)##.

3. Calculating, we get AB = ##√(9^2 + 12^2)## = √(81 + 144) = √225 = 15 meters.

4. Therefore, the height of the tree before it broke was BC + AB = 9 + 15 = 24 meters.

直角三角形; 线段; 点

Definition of Right Triangle

<note>In the figure of this problem, in triangle ABC, angle BCA is a right angle (90 degrees)</note>, therefore triangle ABC is a right triangle. <note>Side BC and side CA are the legs</note>, <note>side AB 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 <note>ABC</note>, <note>angle BCA is a right angle (90 degrees)</note>, <note>sides BC and AC are the legs</note>, <note>side AB is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>AB^2 = BC^2 + AC^2</note>.


Question: As shown in the figure, the rectangle ABCD is folded along the segment OG to the position OB'C'G, ∠OGC' equals 100°, then the degree of ∠DGC' is ()

Answer: 20°

1. Given that rectangle ABCD is folded along segment OG to the position OB'C'G, ∠OGC' equals 100°.

2. Due to the symmetry in the folding process, ∠OGC equals ∠OGC', i.e., ∠OGC = 100°.

3. According to the properties of a straight angle, ∠OGD = 180° - ∠OGC.

4. Substituting the value of ∠OGC, we get ∠OGD = 180° - 100° = 80°.

5. ####∠DGC' = ∠OGC' - ∠OGD.

6. Substituting the known values, we get ∠DGC' = 100° - 80° = 20°.

7. Through the above reasoning, the final answer is 20°.

反射; 邻补角; 矩形; 普通三角形

Definition of Linear Pair of Angles

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

<note>∠OGD and ∠OGC share a common side OG</note>, <note>their other sides OD and OC are extensions of each other in opposite directions</note>, so <note>∠OGD and ∠OGC are adjacent supplementary angles</note>.

Definition of Straight Angle

<note>Ray GC rotates around endpoint G to form a straight line with the initial side, creating straight angle CGD</note>. According to the definition of a straight angle, the measure of a straight angle is 180 degrees, i.e., <note>angle CGD = 180 degrees</note>.


Question: Given: AB∥CD, ∠ABE=120°, ∠C=25°, find the degree measure of ∠α.

Answer: 85°

1. Draw line EF through point E, making EF∥CD.

2. According to the parallel axiom 2 of parallel lines, alternate interior angles are equal, so ∠CEF = ∠ECD = 25°.

3. Since AB∥CD, according to the transitivity of parallel lines, we get EF∥AB.

4. According to the parallel axiom 2 of parallel lines, the same side interior angles are supplementary, therefore ∠ABE + ∠BEF = 180°.

6. From the above steps, we get ∠α = ∠BEF + ∠FEC = 60° + 25° = 85°.

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

平行线; 同位角; 内错角; 三角形的外角

Definition of Alternate Interior Angles

In the figure of this problem, <note>two parallel lines CD and EF</note> are intersected by <note>line CE</note>, where <note>∠CEF and ∠ECD are located between the two parallel lines and on opposite sides of the transversal CE</note>, thus <note>∠CEF and ∠ECD are alternate interior angles</note>. <note>Alternate interior angles are equal, i.e., ∠CEF=∠ECD=25°</note>.

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

<note>In the diagram of this problem, two lines AB and EF are intersected by a third line BE</note>, <note>the two angles ∠ABE and ∠BEF are on the same side of the intersecting line BE, and within the intersected lines AB and EF</note>, so <note>∠ABE and ∠BEF are consecutive interior angles</note>. <note>Consecutive interior angles ∠ABE and ∠BEF are supplementary, that is, angle ABE + angle BEF = 180 degrees</note>.

Parallel Postulate 2 of Parallel Lines

In the diagram of this problem, two parallel lines <note>AB</note> and <note>CD</note> are intersected by a third line <note>BE</note>, forming the following geometric relationships:
1. Corresponding angles: <note>∠ABE</note> and <note>∠FED</note> are equal.
2. Alternate interior angles: <note>∠ECD</note> and <note>∠FEC</note> are equal.
3. Consecutive interior angles: <note>∠ABE</note> and <note>∠BEF</note> are supplementary, that is <note>∠ABE + ∠BEF = 180 degrees</note>.
These relationships illustrate that when two parallel lines are intersected by a third line, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.

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


Question: As shown in the figure, two vertices of a right triangle with a 30° angle are placed on opposite sides of a rectangle. If ∠1 = 25°, then the measure of ∠2 is ()

Answer: 115°

1. Given quadrilateral ABCD is a rectangle ##ABCD counterclockwise takes the vertices of the rectangle in order, A is the top left vertex. The right triangle intersects BC and AD at F and E respectively, the other angle is set as G. According to the definition of the rectangle##, it is concluded that AD is parallel to BC.

2. ∠2 and ∠DEF are between two parallel lines AD and BC, according to the parallel axiom 2 of parallel lines, it is obtained that ∠2 is equal to ∠DEF.

3. Given ∠1=25°, through the algebraic relationship of angles, set the auxiliary angle ∠GEF equal to 90°.

4. According to the angle relationship in triangle DEF, it is obtained that ∠DEF = ##∠DEG＋∠GEF##= 25° + 90°.

5. ####it is obtained that ∠2 = 115°.

6. Through the above reasoning, the final answer is 115°

直角三角形; 矩形; 邻补角; 三角形的外角; 平行线

Definition of Rectangle

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

In the figure of this problem, the quadrilateral <note>ABCD</note> is a <note>rectangle</note>, its interior angles <note>∠A, ∠B, ∠C, ∠D</note> are all right angles (90 degrees), and <note>side AD is parallel and equal in length to side BC</note>, <note>side AB is parallel and equal in length to side CD</note>.

Definition of Alternate Interior Angles

In the figure of this problem, <note>two parallel lines AD and BC</note> are intersected by a line <note>DF</note>, where <note>angle ∠2 and angle ∠DEF</note> are located between the two parallel lines and on <note>opposite sides of the intersecting line DF</note>. Therefore, <note>angle ∠2 and angle ∠DEF are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>angle ∠2 is equal to angle ∠DEF</note>.

Parallel Postulate 2 of Parallel Lines

<note>AD is parallel to BC</note>, the line <note>DEF</note> intersects to form alternate interior angles <note>∠2 equals ∠DEF</note>. According to the theorem of alternate interior angles of parallel lines, <note>AD is parallel to BC</note>, the line <note>DEF</note> intersects to form alternate interior angles <note>∠2 equals ∠DEF</note>.

Triangle Angle Sum Theorem

In the diagram of this problem, <note>in triangle GEF, angles GEF, GFE, and FGE are the three interior angles of triangle GEF</note>. According to the Triangle Angle Sum Theorem, <note>the sum of the three</note> is 180°.

Definition of Right Triangle

In the diagram for this problem, <note>triangle GEF</note> has <note>angle GEF as a right angle (90 degrees)</note>, therefore <note>triangle GEF</note> is a right triangle. <note>Side GE and side EF are the legs</note>, <note>side GF is the hypotenuse</note>.


Question: As shown in the figure, in △ABC, AB=2, AC=1, the circle with AB as its diameter is tangent to AC and intersects side BC at point D. Find the length of AD.

Answer: \frac{2}{5}√{5}

1. Since the circle with diameter AB is tangent to AC, ∠CAB=90°. Because AB is the radius, ∠ADB is the inscribed angle opposite the diameter. According to the inscribed angle theorem, the inscribed angle opposite the diameter is a right angle, so ∠ADB=90°.

2. Given AB=2, AC=1. In right triangle ABC, according to the Pythagorean theorem, we get BC=√(AB^2 + AC^2)=√(2^2 + 1^2) = √5.

3. In triangle ADB, ∠DAB+∠DBA=180°-90°=90°. Given ∠CAB=90°=∠CAD+∠DAB, we can deduce that ∠DBA=∠CAD. According to the similarity criterion (AA), triangle ADB is similar to triangle CAB.

4. According to the definition of similar triangles, we get: AD/AB = AC/BC.

5. Based on the previous conclusion: AD = AC * AB / BC = 1 * 2 / √5 = 2 / √5, simplified to AD = (2√5) / 5.

圆; 切线; 圆周角; 直角三角形; 弦

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.

In the circle with <note>diameter AB</note>, <note>point A is the tangent point of line AC and the circle</note>, and <note>line segment AB is the diameter of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent line AC is perpendicular to the diameter BA passing through the tangent point A</note>, that is, <note>∠CAB = 90 degrees</note>.

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

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

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

The angle subtended by the diameter <note>AB in a circle</note> is a right angle (90 degrees) <note>ADB</note>.

Pythagorean Theorem

In the <note>right triangle ABC</note>, <note>angle ∠CAB is a right angle (90 degrees)</note>, <note>sides AC and AB are the legs</note>, <note>side BC is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>BC^2 = AB^2 + AC^2</note>, that is, <note>BC = √{AB^2 + AC^2} = √{2^2 + 1^2} = √5</note>.

Similarity Theorem for Triangles (AA)

In the diagram of this problem, in triangles ADB and ABC, <note>angle DAB is equal to angle ACD</note>, and triangles ADB and ABC <note>share ∠B</note>, so triangle ADB <note>is similar to triangle ABC</note>.

Definition of Similar Triangles

<note>Triangle ADB and triangle ABC are similar triangles</note>. According to the definition of similar triangles: <note>∠ADB = ∠CAB, ∠DAB = ∠ACB, ∠DBA = ∠ABC</note>; <note>AC/BC = AD/AB</note>.

Definition of Inscribed Angle

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

<note>In the circle with AB as the diameter</note>, the vertex of angle ADB <note>(point D)</note> is on the circumference, and the two sides of angle ADB intersect the circle at <note>points A and B</note>. Therefore, <note>angle ADB is an inscribed angle</note>.


Question: Given that line a ∥ b, ∠1 and ∠2 are complementary, ∠3 = 121°, then ∠4 equals ()

Answer: 149°

1. Given that line a∥b, ##let the adjacent supplementary angle of ∠2 be ∠5, the common side of ∠2 and ∠5 is c, the other side is a, and they are opposite extensions##, we get ∠2 + ∠5 = 180°.

2. Given ∠3 = 121°, ##according to the parallel axiom 2 of parallel lines, ∠3 and ∠5 are corresponding angles##, so ∠5 = ∠3 = 121°.

3. From the above result, we get ∠2 = 180° - 121° = 59°.

4. According to the problem statement, ∠2 and ∠1 are complementary, so ∠2 + ∠1 = 90°, thus ∠1 = 90° - 59° = 31°.

5. Since line a∥b, ##according to the parallel axiom 2 of parallel lines, ∠1 and ∠4 are interior angles on the same side##, we get ∠1 + ∠4 = 180°.

6. Given ∠1 = 31°, so ∠4 = 180° - 31° = 149°.

7. Through the above reasoning, the final answer is 149°.

平行线; 同位角; 内错角; 邻补角; 对顶角

Parallel Postulate 2 of Parallel Lines

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

<note>In the diagram of this problem, two parallel lines a and b are intersected by a third line cd</note>, forming the following geometric relationships: 1. Corresponding angles: <note>angle 3 and angle 5</note> are equal. 2. Consecutive interior angles: <note>angle 1 and angle 4</note> are supplementary, that is, <note>angle 1 + angle 4 = 180 degrees</note>. These relationships illustrate that when two parallel lines are intersected by a third line, <note>corresponding angles are equal</note>, <note>consecutive interior angles are supplementary</note>.

Definition of Corresponding Angles

In the diagram of this problem, <note>two parallel lines a and b are intersected by a line c</note>, <note>where angle 3 and angle 5 are on the same side of the intersecting line c and on the same side of the intersected lines a and b</note>, thus <note>angle 3 and angle 5 are corresponding angles</note>. Corresponding angles are equal, that is, <note>angle 3 is equal to angle 5</note>.

In the figure of this problem, <note>two lines a and b are intersected by a third line d</note>, <note>angles 1 and 4 are on the same side of the transversal d, and within the intersected lines a and b</note>, so <note>angles 1 and 4 are consecutive interior angles</note>. <note>Consecutive interior angles 1 and 4 are supplementary, that is, angle 1 + angle 4 = 180 degrees</note>.


Question: As shown in the figure, line a ∥ b, ∠1 = 72°, then the degree of ∠2 is ()

Answer: 108°

1. Given that line a ∥ b, ##let the adjacent supplementary angle of ∠2 above line b be ∠3. According to the parallel axiom 2 of parallel lines##, we get ∠1 = ∠3 (corresponding angles are equal).

2. Given ∠1 = 72°, so according to step 1, we get ∠3 = 72°.

3. According to ##the definition of adjacent supplementary angles##, we get ∠3 + ∠2 = 180°.

4. #### ∠3 = 72° substituted into the equation in step 3, we get ∠2 = ##180°-∠3=##180° - 72° = 108°.

5. Through the above reasoning, we finally get the degree of ∠2 as 108°.

The original text: <note>Line a ∥ b</note>, intersected by <note>line c</note>, where <note>∠1 and ∠3 on the same side of line c are corresponding angles</note>. According to the definition, <note>corresponding angles are equal</note>, so <note>∠1 = ∠3</note>.

Parallel Postulate 2 of Parallel Lines

Original: <note>Two parallel lines a and b are intersected by a third line c</note>, forming the following geometric relationships:
1. Corresponding angles: <note>Angle 1 and Angle 3 are equal</note>.
2. Alternate interior angles: None.
3. Same-side interior angles: None.
These relationships indicate that when two parallel lines are intersected by a third line, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary.

Definition of Linear Pair of Angles

<note>In the diagram of this problem, ##∠3## and ####∠2 share a common side c</note>, their other sides <note>a and b are extensions in opposite directions</note>, so <note>##∠3## and ∠2 are adjacent supplementary angles</note>.


Question: As shown in the figure, in the right triangle ABC, ∠C=90°, AB=5, AC=4, find the value of sin∠B.

Answer: \frac{4}{5}

1. Given in the right triangle ABC, ∠C = 90°, AB = 5, AC = 4.

2. ##According to the definition of the sine function##, sin∠B = opposite side/hypotenuse.

3. In this problem, the opposite side of angle B is AC, and the hypotenuse is AB.

4. Substituting the given conditions, ##sin∠ABC## = AC / AB = 4 / 5.

5. Through the above reasoning, the final value of sin∠B is 4/5.

Definition of Right Triangle

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

Definition of Sine Function

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

<note>In the figure of this problem, in the right triangle ABC, ##∠ABC## is an acute angle</note>, side AC is <note>the opposite side of ##∠ABC##</note>, side AB is <note>the hypotenuse</note>. According to the definition of the sine function, <note>the sine value of ##∠ABC## is equal to the ratio of the opposite side AC to the hypotenuse AB</note>, that is, <note>##sin(∠ABC)## = AC / AB = 4 / 5</note>.

Hypotenuse of a Right Triangle

In a right triangle, the hypotenuse is the side opposite the right angle, and it is also the longest side among the three sides of the triangle.

In the figure of this problem, for angle ∠BAC, BC is the opposite side, for angle ∠B, AC is also the opposite side, and AB is the hypotenuse.


Question: As shown in the figure, AB is the diameter of ⊙O, and points C and D are on ⊙O. If ∠ABD=50°, then the degree of ∠BCD is ()

Answer: 40°

1. Connect AD, according to the problem statement: AB is the diameter of ⊙O.

2. According to ##(Corollary 2 of the Inscribed Angle Theorem)##, the inscribed angle subtended by the diameter is a right angle, thus ∠ADB equals 90°.

3. Given ∠ABD equals 50°, by the sum of the interior angles in a triangle, in △ADB, we can obtain ∠DAB=##180°-∠ADB-∠ABD=180°-##90°-50°=40°.

4. Since AC and AD are chords of the same circle, ##according to Corollary 1 of the Inscribed Angle Theorem, in a circle, any two inscribed angles subtended by the same arc are equal##, we can obtain ∠BCD=∠DAB=40°.

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

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 <note>the center O</note> and <note>points A and B on the circumference</note>, with a length of <note>2 times the radius, i.e., AB = 2 * OA</note>.

Definition of Inscribed Angle

<note>In the figure of this problem, in circle O, the vertex D of angle ∠ADB is on the circumference, and the two sides of angle ∠ADB intersect circle O at points A and B respectively. Therefore, angle ∠ADB is an inscribed angle.</note>
<note>In circle O, the vertex B of angle ∠ABD is on the circumference, and the two sides of angle ∠ABD intersect circle O at points A and D respectively. Therefore, angle ∠ABD is an inscribed angle.</note>
<note>In circle O, the vertex A of angle ∠DAB is on the circumference, and the two sides of angle ∠DAB intersect circle O at points D and B respectively. Therefore, angle ∠DAB is an inscribed angle.</note>
<note>In circle O, the vertex C of angle ∠BCD is on the circumference, and the two sides of angle ∠BCD intersect circle O at points B and D respectively. Therefore, angle ∠BCD is an inscribed angle.</note>

Original: <note>In the diagram of this problem, in circle O, diameter AB</note> subtends a <note>right angle (90 degrees) at the circumference, angle ADB</note>. (Or <note>in the diagram of this problem, the angle ADB at the circumference is 90 degrees</note>, so the chord <note>AB subtended by it is the diameter</note>.)

Triangle Angle Sum Theorem

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

In the figure of this problem, in triangle ADB, <note>angle ABD</note>, <note>angle ADB</note>, and <note>angle DAB</note> are the three interior angles of triangle <note>ADB</note>. According to the <note>Triangle Angle Sum Theorem</note>, angle ABD + angle ADB + angle DAB = 180°.

In a circle, any two inscribed angles that subtend the same arc are equal.

<note>In circle O, the inscribed angles ∠BAD and ∠BCD corresponding to arc BD are equal</note>. According to Corollary 1 of the Inscribed Angle Theorem, <note>the inscribed angles ∠BAD and ∠BCD corresponding to the same arc BD are equal, that is, ∠BAD = ∠BCD</note>.


Question: As shown in the figure, C and D are points on the line segment AB. If AC=3cm, C is the midpoint of AD, and AB=10cm, then DB=()

Answer: 4cm

1. Given that point C is the midpoint of segment AD, and AC=3cm, according to the definition of the midpoint of a segment, we get CD=3cm.

2. From the given segment AB=10cm, we can write the equation AC + CD + DB = AB.

3. Substituting the values, we get 3cm + 3cm + DB = 10cm.

4. Solving the equation, we get DB = 10cm - 3cm - 3cm = 4cm.

5. Through the above reasoning, the final answer is 4cm.

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 AD is point C</note>. According to the definition of the midpoint of a line segment, <note>point C divides line segment AD into two equal parts</note>, that is, <note>the lengths of line segment AC and line segment CD are equal</note>. That is, <note>AC = CD = 3cm</note>.


Question: As shown in the figure, ∠A=70°, ∠2=130°, then ∠1=()

Answer: 120°

1. Given angle A = 70°, ∠2 = 130°.

####

2. According to the exterior angle property of triangle ABC: the exterior angle is equal to the sum of the two non-adjacent interior angles. Therefore, ∠ABC = ∠2 - ∠A.

3. Substituting the given data: ∠CAB = 70°, ∠2 = 130°, then angle ∠ABC = 130° - 70° = 60°.

4. Let there be a point D on the extension of CB, angle ∠1 and ∠ABC are supplementary adjacent angles. According to the definition of supplementary adjacent angles, ∠1 = 180° - ∠ABC = 180° - 60° = 120°.

Exterior Angle Theorem of Triangle

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

<note>In the figure of this problem, angle 2 is an exterior angle of triangle ABC</note>, angle ∠CAB and angle <note>∠ABC are the two interior angles not adjacent to exterior angle ∠2</note>, according to the Exterior Angle Theorem of Triangle, <note>exterior angle ∠2 is equal to the sum of the two non-adjacent interior angles ∠CAB and ∠ABC</note>, that is angle <note>∠1 = ∠CAB + ∠ABC</note>.

Definition of Linear Pair of Angles

<note>In the figure of this problem, ## ∠1 and ∠ABC ##</note> have a common <note>side AB</note>, and their other sides <note>BD and BC</note> are extensions in opposite directions, so <note>##∠1 and ∠ABC## are adjacent supplementary angles.</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.

The original text: <note>One interior angle of polygon ABC is ∠ACB</note>, extending the adjacent sides of this interior angle <note>BC and AC forming angle ∠2</note> is called the exterior angle of the interior angle <note>∠ACB</note>.


Question: As shown in the figure, line a ∥ b, line c intersects lines a and b at points A and B respectively, AM ⊥ b, and the foot of the perpendicular is point M. If ∠1 = 58°, then ∠2 = ()

Answer: 32°

1. Given that line a∥b, tangent c intersects lines a and b at points A and B respectively, and ∠1=58°.

2. According to ##Parallel Lines Postulate 2, corresponding angles are equal##, thus ∠ABM=∠1=58°.

3. According to the problem statement, AM is perpendicular to line b, with the foot of the perpendicular at point M, ##according to the definition of perpendicular lines##, this means ∠AMB=90°.

4. According to the Triangle Sum Theorem, the sum of the three interior angles of triangle ABM is 180°.

5. In triangle ABM, according to the ##Triangle Sum Theorem##, it can be expressed as: ∠2+∠ABM+∠AMB=180°.

6. Substitute the angle values: ∠2+58°+90°=180°.

7. Simplify the equation to get: ∠2=180°-58°-90°=32°.

8. Through the above reasoning, the final answer is 32°.

平行线; 同位角; 内错角; 垂线; 直角三角形

Definition of Corresponding Angles

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

<note>In the figure of this problem, line a ∥ b, line c is the transversal</note>. ##<note>∠1</note>## and <note>∠ABM</note> are corresponding angles. According to the properties of parallel lines, <note>∠ABM=##∠1##=58°</note>.

Definition of Foot of a Perpendicular

The intersection point of a perpendicular line with the segment it is perpendicular to is called the foot of the perpendicular.

<note>Line AM intersects line b at point M</note>, and <note>line AM is perpendicular to line b</note>, therefore <note>point M is the foot of the perpendicular from line AM to line b</note>.

Triangle Angle Sum Theorem

<note>Triangle ABM</note> has interior angles <note>∠ABM</note>, <note>∠AMB</note>, and <note>∠BAM</note>. According to the <note>Triangle Angle Sum Theorem</note>, <note>∠ABM + ∠AMB + ∠BAM = 180°</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>The angle ∠AMB formed by the intersection of line AM and line BM is 90 degrees</note>, therefore according to the definition of perpendicular lines, <note>line AM and line BM are perpendicular to each other</note>.

Parallel Postulate 2 of Parallel Lines

In the figure of this problem, <note>two parallel lines a and b are intersected by a third line c</note>, forming the following geometric relationship: 1. Corresponding angles: <note>angle 1 is equal to angle ABM</note>. This relationship indicates that when two parallel lines are intersected by a third line, the corresponding angles are equal.


Question: As shown in the figure, AB is the diameter of ⊙O, point C is on the extension line of AB, CD is tangent to ⊙O at point D. If ∠A = 35°, then ∠C equals ()

Answer: 20°

1. According to the problem statement, connect BD. Because AB is the diameter of ⊙O, ##according to (Theorem 2 of the Inscribed Angle Theorem) the inscribed angle subtended by the diameter is a right angle,## so ∠ADB is a right angle, i.e., ∠ADB=90°.

 2. Given ∠A=35° and ∠ADB=90°, according to the triangle angle sum theorem, we can get ∠ABD=##180°-##90°-∠A=55°.

 3. Since CD is a tangent line and the point of tangency is D, according to the property of the tangent line, ##∠ODC=90°##.

 ##4. Because OA and OD are radii of the circle, therefore OA=OD, according to the definition of an isosceles triangle, triangle OAD is an isosceles triangle. According to the property of an isosceles triangle, ∠A=∠ADO=35°.##

 ##5. Based on the above conclusions, given ∠A=35°, ∠ADC=∠ADO+∠ODC=35°+90°=125°, so according to the triangle angle sum theorem, we can get ∠ACD=180°-∠ADC-∠A=180°-125°-35°=20°.##

 ##6##. Through the above reasoning, the final answer is 20°.

圆; 切线; 圆周角; 直角三角形; 内错角

Definition of Diameter

In this problem diagram, <note>AB is the diameter of circle O</note>, connecting <note>the center O and points A and B on the circumference</note>, with a length of <note>2 times the radius, i.e., AB = 2r</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.

<note>The circle ⊙O and the line CD have only one common point D</note>, <note>this common point is called the point of tangency</note>. Therefore, <note>the line CD is the tangent to the circle ⊙O</note>.

Triangle Angle Sum Theorem

In the figure of this problem, in triangle ABD, <note>∠BAD</note>, <note>∠ABD</note>, and <note>∠ADB</note> are the three interior angles of triangle ABD. According to the Triangle Angle Sum Theorem, <note>∠BAD + ∠ABD + ∠ADB = 180°</note>. Similarly, in triangle ADC, <note>∠ADC</note>, <note>∠DAC</note>, and <note>∠ACD</note> are the three interior angles of triangle ADC. According to the Triangle Angle Sum Theorem, <note>∠ADC + ∠DAC + ∠ACD = 180°</note>.

<note>In the diagram of this problem, in circle O, point D is the point of tangency where line CD touches the circle</note>, <note>segment OD is the radius of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent line CD is perpendicular to the radius OD at the point of tangency D</note>, that is, <note>∠ODC=90°</note>.

<note>In the diagram of this problem, in circle O, the angle subtended by the diameter AB at the circumference, angle ADB, is a right angle (90 degrees)</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.

In the <note>isosceles triangle ADO</note>, <note>sides OA and OD are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, i.e., <note>angle ADO = angle OAD</note>.

Definition of Isosceles Triangle

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

In the figure of this problem, in triangle ADO, <note>side OA and side OD are equal</note>, therefore <note>triangle ADO is an isosceles triangle</note>.


Question: As shown in the figure, in parallelogram ABCD, AC and BD are diagonals, BC=6, the height on side BC is 4, then the area of the shaded part in the figure is ()

Answer: 12

1. Given that the diagonals AC and BD of parallelogram ABCD intersect at point O, and BC=6, the height on side BC is 4. ##Area of the parallelogram = 4*6=24##

2. According to the properties of a parallelogram, the diagonals divide the parallelogram into four triangles of equal area.

####

##3. Since the parallelogram is a centrally symmetric figure, based on the properties of central symmetry, we can deduce that the two shaded triangles in the upper part of the parallelogram are congruent to the two blank triangles in the lower part. Moreover, since the diagonals AC and BD divide the parallelogram into four triangles of equal area, the shaded part occupies the area of two of these triangles. Therefore, the area of the shaded part##=2/4*Area of parallelogram ABCD=12 square units.##

##4. ##Based on the above reasoning, the final answer is 12.

平行四边形; 对称; 线段; 垂线; 普通三角形

Definition of Parallelogram

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

<note>Quadrilateral ABCD 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 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 perpendicular to the opposite side AD (or its extension) from vertex B</note> is the altitude from vertex B. <note>The line segment forms a right angle (90 degrees) with the side AD (or its extension)</note>, indicating that the line segment is the perpendicular distance from vertex B to the opposite side AD (or its extension).

Definition of Diagonal

A diagonal is a line segment connecting one vertex of a polygon to another vertex that is not adjacent to it.

In the figure of this problem, in parallelogram ABCD, vertices A, B, C, D, <note>the diagonal is the line segment AC connecting vertex A and the non-adjacent vertex C</note>, <note>and the line segment BD connecting vertex B and the non-adjacent vertex D</note>. Therefore, <note>the line segments AC and BD are the diagonals of parallelogram ABCD</note>.

Area Formula of a Parallelogram

The area of a parallelogram is calculated by multiplying the length of its base by the corresponding height, which can be expressed as \( A = \text{base length} \times \text{height} \).

<note>In parallelogram ABCD, side BC is the base</note>, <note>the corresponding height is the perpendicular distance from base BC to the opposite side AD, denoted as 4</note>. Therefore, according to the area formula of a parallelogram, the area of parallelogram ABCD is equal to <note>the length of base BC multiplied by the corresponding height 4</note>, that is, <note>S_{ABCD} = BC × height = 6 × 4 = 24</note>.

Properties of Parallelogram Theorem

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

In the diagram 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, that is, the intersection point O divides diagonal AC into two equal segments AO and OC</note>, <note>and divides diagonal BD into two equal segments BO and OD</note>.

Definition of Central Symmetric Figure

A figure in the plane is called a central symmetric figure if it coincides with itself when rotated 180° about a certain point. This particular point is referred to as its center of symmetry.

<note>After rotating the figure ABCD 180° around point O, it completely coincides with the original figure</note>. According to the definition of a central symmetric figure, <note>ABCD is a central symmetric figure, and point O is its center of symmetry</note>.

Definition of Central Symmetry

In a plane, if a shape can be rotated 180° around a certain point and the resulting shape coincides with another shape, then the two shapes are said to be centrally symmetric about this point. This point is called the center of symmetry. The points that coincide after rotation by 180° are called symmetric points.

Original: <note>If the two shaded triangles in the upper half of the parallelogram are rotated 180° around point O</note> and completely overlap with <note>the two blank triangles in the lower half</note>, then according to the definition of central symmetry, <note>they are pairwise centrally symmetric about point O</note>, <note>point O is their center of symmetry</note>, and the corresponding points are called <note>symmetric points</note>.

Properties of Central Symmetry

A geometric figure exhibits central symmetry if and only if for every pair of symmetric points, the line segment connecting them passes through the center of symmetry and is bisected by it.

In the figure of this problem, in the centrally symmetric figure <note>ABCD</note>, each line passing through <note>point O</note> intersects the sides of the parallelogram at two points that are a pair of symmetric points. According to the properties of <note>central symmetry</note>, the line segment connecting each pair of symmetric points in the centrally symmetric figure passes through the <note>center of symmetry O</note> and is bisected by the <note>center of symmetry O</note>.


Question: As shown in the figure, in △ABC, DE∥BC, intersecting AB and AC at points D and E respectively. If AD=2, DB=3, BC=6, then the length of DE is ()

Answer: \frac{12}{5}

1. Given AD=2, DB=3, using segment addition, we get AB=AD+DB=2+3=5.

2. Because DE∥BC, ##according to the parallel axiom 2 of parallel lines, corresponding angles are equal, that is ∠ADE=∠ABC, and ∠BAC and ∠DAC are the same angle, according to the similarity theorem (AA), △ADE ∽ △ABC##.

3. According to the ##definition## of similar triangles, in this problem it is expressed as: rac{AD}{AB} = rac{DE}{BC}.

4. Substitute the known side lengths into the proportion, we get rac{2}{5} = rac{DE}{6}.

5. Solve for DE by cross-multiplication, we get DE = rac{2 	imes 6}{5} = rac{12}{5}.

6. Through the above reasoning, the final answer is rac{12}{5}.

平行线; 普通三角形; 线段; 内错角; 等腰三角形

Similarity Theorem for Triangles (AA)

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

In <note>triangle ADE</note> and <note>triangle ABC</note>, <note>angle ADE is equal to angle ABC</note>, and <note>angle DAE is equal to angle BAC</note>, so <note>triangle ADE is similar to triangle ABC</note>.

Definition of Corresponding Angles

<note>Two parallel lines DE and BC</note> are intersected by <note>a line AB</note>, where <note>angle ADE and angle ABC are on the same side of the intersecting line AB, and on the same side of the intersected lines DE and BC</note>. Therefore, <note>angle ADE and angle ABC are corresponding angles</note>. Corresponding angles are equal, i.e., <note>angle ADE is equal to angle ABC</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 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>\frac{AD}{AB} = \frac{DE}{BC} = \frac{AE}{AC}</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</note> are intersected by <note>a third line AB</note>, forming the following geometric relationship: <note>Corresponding angles: angle ADE and angle ABC are equal</note>.


Question: As shown in the figure, in △ABC, DE∥BC, if AD=3, DB=6, DE=2.5, then the length of BC is ()

Answer: 7.5

1. Given DE∥BC, according to the parallel postulate 2 of parallel lines, corresponding angles are equal, it can be known that ∠ADE=∠B, ∠AED=∠C. According to the similarity theorem of triangles (AA), △ADE ∽ △ABC.

2. By the definition of similar triangles, it can be derived that AD/AB = DE/BC.

3. Given AD=3, DB=6, then AB=AD+DB=3+6=9.

4. Substitute the known values of AB and DE into the proportion, we get 3/9 = 2.5/BC.

5. By solving the proportion, we can find BC, that is BC = 2.5×{9/3} = 7.5.

6. Through the above reasoning, the final answer is 7.5.

<note>Two parallel lines DE and BC are intersected by a third line AB and line AC</note>, forming the following geometric relationships:
Corresponding angles: <note>∠ADE=∠B</note>, <note>∠AED=∠C</note>.
These relationships indicate that when two parallel lines are intersected by a third line, the corresponding angles are equal.

Definition of Corresponding Angles

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

<note>DE and BC are intercepted by the third line AB and line AC</note>, where <note>∠ADE and ∠B</note>, <note>∠AED and ∠C</note> are on the same side of the intercepting lines AB and AC, respectively, and on the same side of the intercepted lines DE and BC. Therefore, <note>∠ADE and ∠B</note>, <note>∠AED and ∠C are corresponding angles</note>. Corresponding angles are equal, that is, <note>∠ADE=∠B</note>, <note>∠AED=∠C</note>.

Similarity Theorem for Triangles (AA)

In the diagram of this problem, triangle ADE and triangle ABC, because ∠ADE=∠B, ∠AED=∠C, therefore triangle ADE and triangle ABC are similar.

Definition of Similar Triangles

In the figure of this problem, <note>triangle ADE and triangle ABC are similar triangles</note>. According to the <note>definition of similar triangles</note>, we have: <note>AD/AB = DE/BC</note>.


Question: As shown in the figure, AB∥CD, AB=6, CD=9, AD=10, find the length of OD ()

Answer: 6

1. Given AB∥CD, according to the ##parallel lines axiom 2, alternate interior angles are equal##, we get ∠BAO = ##∠CDO##, and ∠ABO = ##∠DCO##.

2. Since two interior angles of the two triangles are respectively equal, according to the ##similar triangles criterion theorem (AA)##, we get △AOB∽△DOC.

3. From the ##definition of similar figures##, we get AB/CD = AO/OD.

4. Based on the given conditions, substitute the lengths AB=6, CD=9, AD=10, we get 6/9 = (AD-OD)/OD.

5. Simplifying the proportion, we get 6/9 = (10-OD)/OD.

6. By solving the equation, we get OD = 6.

7. Through the above reasoning, we finally get the answer as 6.

平行线; 内错角; 位似

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 AB and CD</note> are intersected by <note>line AD and line BC</note>, where <note>angle A and angle D are located between the two parallel lines</note>, <note>angle B and angle C are located between the two parallel lines</note>, and respectively on <note>opposite sides of the intersecting lines AD and BC</note>. Therefore, <note>angle A and angle D are alternate interior angles</note>, <note>angle B and angle C are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>angle A is equal to angle D</note>, <note>angle B is equal to angle C</note>.

Parallel Postulate 2 of Parallel Lines

<note>AB∥CD</note>, and it is intersected by the lines <note>AD and BC</note>, forming the following geometric relationships: <note>alternate interior angles ∠BAO and ∠CDO are equal, ∠ABO and ∠DCO are equal</note>.

Similarity Theorem for Triangles (AA)

In <note>triangles AOD and COD</note>, if <note>∠BAO = ∠CDO</note> and <note>∠ABO = ∠DCO</note>, then <note>triangle AOD is similar to triangle COD</note>.

Definition of Similar Triangles

<note>In the figure of this problem, triangle AOD and triangle COD are similar figures</note>. According to the definition of similar triangles: <note>angle A = angle D</note>, <note>angle B = angle C</note>, <note>angle AOB = angle COD</note>; <note>OA/OD = OB/OC = AB/CD</note>.


Question: As shown in the figure, given that lines AB and CD intersect at point O, OE ⊥ AB, ∠EOC = 30°, then the degree of ∠BOD is ()

Answer: 120°

1. Given that lines AB and CD intersect at point O, and OE is perpendicular to AB, ##according to the definition of perpendicular lines, we have ∠EOB## = 90°.

####

##2##. Since ∠EOC = 30°, and ##∠EOB## = 90°, therefore ∠COB = 90° - 30° = 60°.

##3##. According to the definition of a straight angle, lines AB and CD intersect at point O, forming the ##straight angle ∠COD, and ∠COD## = ∠COB + ∠BOD.

##4##. From the definition of a straight angle, we know that ##∠COD## = 180°, thus we have 180° = ∠COB + ∠BOD.

##5##. Since ∠COB = 60°, therefore ∠BOD = 180° - 60° = 120°.

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

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.

In the figure of this problem, ray <note>CO rotates around endpoint O to form a straight line with the initial side, creating a straight angle COD</note>. According to the definition of a straight angle, <note>a straight angle measures 180 degrees, i.e., angle COD = 180 degrees</note>.

Definition of Linear Pair of Angles

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

<note>Angle COB and angle DOB share a common side OB</note>, their other sides <note>OC and OD are extensions in opposite directions</note>, so <note>angle COB and angle DOB are adjacent supplementary angles</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 EO and line AB are perpendicular to each other</note>, therefore according to the <note>definition of perpendicular lines</note>, the angle formed by the intersection of line EO and line AB, <note>∠EOB, is 90 degrees</note>.


Question: As shown in the figure, in ⊙O, AC is the diameter, MA and MB are tangent to ⊙O at points A and B respectively, ∠BAC=25°, then the measure of ∠AMB is ()

Answer: 50°

1. Given that AC is the diameter and MA is tangent to ⊙O at point A, according to the ##property of the tangent line of a circle##, we get ∠MAC=90°.

 2. Given ∠BAC=25°, ##therefore## ∠MAB=∠MAC-∠BAC=90°-25°=65°.

 3. Since MA and MB are tangent to ⊙O at points A and B respectively, according to the ##theorem of the length of tangents##, we get MA=MB.

 4. According to the property of the isosceles triangle, ∠MAB=∠MBA=65°.

 5. In △MAB, according to the triangle angle sum theorem, we get ∠AMB=180°-(∠MAB+∠MBA)=180°-(65°+65°)=50°.

圆; 切线; 圆周角; 等腰三角形; 直角三角形

Definition of an Angle

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

In this problem diagram, <note>∠BAC is a geometric figure formed by two rays BA and CA</note>, <note>these two rays share a common endpoint A</note>. <note>This common endpoint A is called the vertex of angle BAC</note>, and <note>the rays BA and CA are called the sides of angle BAC</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.

In this problem diagram, <note>triangle MAB</note> is a geometric figure composed of three <note>non-collinear points M, A, B</note> and their <note>connecting line segments MA, AB, MB</note>. <note>Points M, A, B are the three vertices of the triangle</note>, and <note>line segments MA, AB, MB are the three sides of the triangle</note>.

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

<note>In circle O, point A is the point of tangency between line MA and the circle</note>, <note>line segment OA is the radius of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent line MA is perpendicular to the radius OA at the point of tangency A</note>, i.e., <note>∠MAC=90°</note>.

Triangle Angle Sum Theorem

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

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

Properties of Isosceles Triangle

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

In the figure of this problem, in the isosceles triangle <note>MAB</note>, side <note>MA</note> and side <note>MB</note> are equal. Therefore, according to the properties of the isosceles triangle, the angles opposite the equal sides are equal, that is, <note>∠MAB = ∠MBA = 65°</note>.

Tangent-Segment Theorem

If two tangent segments are drawn to a circle from an external point, then the lengths of these tangent segments are equal. Moreover, the line segment connecting the center of the circle to the external point bisects the angle formed by the two tangent segments.

<note>In this problem, two tangents MA and MB are drawn from an external point M to the circle, and their tangent lengths are equal, i.e., MA=MB.</note> <note>The line connecting the center of the circle O and the point M bisects the angle between the two tangents MA and MB, i.e., ∠OMA=∠OMB.</note>


Question: As shown in the figure, line m ∥ n, vertex A of Rt△ABC is on line n, ∠C = 90°, AB and CB intersect line m at points D and E respectively, and DB = DE. If ∠B = 25°, then the degree of ∠1 is ()

Answer: 65°

1. As shown in the figure, according to the problem statement, line m is parallel to line n, the vertex A of Rt△ABC is on line n, ∠C=90°, AB and CB intersect line m at points D and E respectively, and DB=DE, ∠B=25°.

2. Because DB=DE, according to the definition of an isosceles triangle: △BDE is an isosceles triangle, ∠B=∠BED (property of isosceles triangles), therefore ∠BED=25°.

3. ∠EDA is an exterior angle of triangle BDE, so according to the exterior angle theorem of triangles: ∠EDA=∠BED+∠B=25°+25°=50°.

4. Let the alternate interior angle of ∠EDA be ∠2, since line m || n, according to the parallel postulate 2, ∠EDA=∠2 (alternate interior angles are equal), therefore ∠2=50°.

5. ∠C=90°, in △ABC, according to the triangle angle sum theorem: ∠BAC + ∠B + ∠C = 180°, then ∠BAC = 180° - ∠B - ∠C = 180° - 25° - 90° = 65°.

6. On line n, because ∠1+∠BAC+∠2=180° (according to the definition of a straight angle), so ∠1=180°-∠BAC-∠2=180°-65°-50°=65°, therefore ∠1 = 65°.

平行线; 直角三角形; 同位角; 内错角; 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.

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

Definition of Parallel Lines

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

<note>Line m and line n lie in the same plane and they do not intersect</note>, therefore, according to the definition of parallel lines, <note>line m and line n are parallel, denoted as m∥n</note>.

Definition of Isosceles Triangle

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

In the figure of this problem, <note>in triangle BDE, side BD and side DE are equal</note>, therefore <note>triangle BDE is an isosceles triangle</note>.

Definition of Alternate Interior Angles

<note>In the figure of this problem, two parallel lines</note> m <note>and</note> n <note>are intersected by a line</note> AB <note>, where</note> ∠ADE <note>and</note> ∠2 <note>are located between the two parallel lines and on opposite sides of the intersecting line</note> AB <note>, thus</note> ∠ADE <note>and</note> ∠2 <note>are alternate interior angles. Alternate interior angles are equal, i.e., angle</note> ∠ADE = ∠2 <note>.</note>

Properties of Isosceles Triangle

<note>In isosceles triangle BDE, sides BD and DE are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, that is, <note>∠DBE = ∠BED = 25°</note>.

Triangle Angle Sum Theorem

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

Parallel Postulate 2 of Parallel Lines

In the diagram of this problem, <note>two parallel lines m and n</note> are intersected by <note>a third line AB</note>, forming the following geometric relationships: 1. Corresponding angles: none. 2. Alternate interior angles: <note>angle EDA and angle 2 are equal</note>. 3. Consecutive interior angles: none. These relationships illustrate that when <note>two parallel lines are intersected by a third line</note>, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.

Definition of Straight Angle

<note>Ray n rotates around endpoint A to form a straight line with the initial side, forming a straight angle</note>. According to the definition of a straight angle, the measure of a straight angle is 180 degrees, that is, <note>angle 1 + angle BAC + angle 2 = 180 degrees</note>.

Exterior Angle Theorem of Triangle

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

In <note>triangle BDE</note>, <note>angle EDA is an exterior angle of the triangle</note>, <note>angle BED and angle B are the two non-adjacent interior angles to the exterior angle EDA</note>. According to the Exterior Angle Theorem of Triangle, <note>the exterior angle EDA is equal to the sum of the two non-adjacent interior angles BED and angle B</note>, that is, <note>angle EDA = angle BED + angle B</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, an interior angle of triangle BED is <note>∠BDE</note>, and the angle <note>∠EDA</note> formed by extending the adjacent sides <note>BD</note> and <note>DE</note> of this interior angle is called the exterior angle of the interior angle <note>∠BDE</note>.