Summary of knowledge points on congruent triangles and related shapes in mathematics for second grade junior high school students published by People's Education Press

Summary of knowledge points of congruent triangles and related graphic knowledge in the second grade of junior high school mathematics published by People's Education Press

Chapter 11

Congruent Triangles Review

Congruent triangles are defined as two triangles that can completely overlap. Congruent triangles have exactly the same shape and size, regardless of position. Through translation, flipping, and rotation, one triangle can be transformed into another congruent triangle.

Congruent triangles have the following properties: corresponding sides are equal, corresponding angles are equal, and do not change due to changes in position.

Understanding: For congruent triangles, the long side corresponds to the long side, and the short side corresponds to the short side. The largest angle corresponds to the largest angle, and the smallest angle corresponds to the smallest angle. Opposite sides of corresponding angles are congruent, and opposite angles of corresponding sides are congruent. Therefore, congruent triangles have equal perimeters and equal areas.

There are three ways to determine congruent triangles: side-side, angle-side, and side-angle-side. Among them, side-side-side (SSS) means that when the three sides of two triangles are equal, the two triangles are congruent. This determination method can be abbreviated as "SSS". In addition, for congruent triangles, the corresponding medians, angle bisectors, and altitudes on corresponding sides are also equal. That is, if two triangles are congruent, then the corresponding medians, angle bisectors, and altitudes on their corresponding sides are also congruent. To sum up,

Side Angle Side: Two sides and their included angles correspond to two triangles that are congruent (SAS). Angles and Sides: Two triangles are congruent (ASA) if their angles and their included sides are equal.

Angular side: Two triangles whose two angles and the opposite side of one of the angles are equal are congruent (can be abbreviated as "AAS") Angular side, hypotenuse, right-angled side

Two right triangles with equal right-angled sides can be proved by the condition that the hypotenuse and the right-angled sides are equal, that is, the "HL" congruence condition. The basic idea of ​​proving that two triangles are congruent is as follows.

): Known two sides (1): Known two sides): Known two sides---Find the third side (SSS) Find the included angle (SAS) Find whether there is a right angle (HL) Find another neighbor here Angle(ASA)

Find another adjacent angle here. Known side and its adjacent angle (2): Known side and angle. Known side and angle - Known side and angle. Known side and its opposite angle. Known angle is a right angle. , find one side

It is known that the angle is a right angle, find one side (HL) Find the other side of this angle (SAS) Find the other side of this angle Find the opposite angle here (AAS) Find one angle (AAS) Find one angle

Find the side between two angles (ASA) Find the side between two angles (3): Known two angles Known two angles - Find any side outside the two angles (AAS) Find outside the two sides Any edge of

2. Bisector of angle: A ray is drawn from the vertex of an angle to divide the angle into two equal angles. This ray is called the bisector of the angle.

1. Properties: The distance from a point on the bisector of an angle to both sides of the angle is equal. 2. Determination: The point that is equidistant from the interior of the angle to both sides of the angle is on the bisector of the angle. When learning congruent triangles, you should pay attention to the following issues:

3. When learning congruent triangles, you should pay attention to the following issues: (1) Correctly distinguish the different meanings of "corresponding sides" and "opposite sides", and "corresponding angles" and "opposite angles"; (2

means that when two triangles are congruent, the letters indicating the corresponding vertices should be written in the corresponding positions; (3) "There are three angles that are equal" or "There are two sides and the opposite angles of one of them are equal." Triangles are not necessarily congruent; (4) Always pay attention to the implicit conditions in graphics, such as "common angles", "common sides", and "opposite angles" (5) Use truncated lengths to supplement short lengths to prove that triangles are congruent.

What are the methods of proving congruent triangles in junior high school

To verify two congruent triangles, generally use side-side (SSS), side-angle side (SAS), angle-side angle (ASA), angle-angle side (AAS), and the hypotenuse and right-angled side (HL) of a right triangle. ) 5 ways to determine.

Measure to judge:

1. SSS (Side-Side-Side): A triangle with three equal sides is a congruent triangle.

2. SAS (Side-Angle-Side): A triangle whose two sides and their included angles are equal is a congruent triangle.

3. ASA (Angle-Side-Angle): Two angles and their included sides are congruent.

4. AAS (Angle-Angle-Side): Two angles and the opposite side of one angle correspond to equal triangles that are congruent.

5. RHS (Right angle-Hypotenuse-Side) (right angle, hypotenuse, side) (also known as HL theorem (hypotenuse, right angle side)): In a pair of right triangles, the hypotenuse and the other right angle side equal. (Its proof is based on the SSS principle)

Extended information:

1. Properties of congruent triangles

1. The corresponding angles of congruent triangles are equal.

2. The corresponding sides of congruent triangles are equal.

3. Vertices that can completely overlap are called corresponding vertices.

4. The heights on the corresponding sides of congruent triangles are equal.

5. The angle bisectors of corresponding angles of congruent triangles are equal.

6. The midlines of corresponding sides of congruent triangles are equal.

7. The area and perimeter of congruent triangles are equal.

8. The values ​​of the trigonometric functions of the corresponding angles of congruent triangles are equal.

2. Inference

1. SSS (Side-Side-Side):

If the lengths of the three sides of each triangle are equal, the two triangles are congruent triangles.

2. SAS (Side-Angle-Side) (side, angle, side):

If the lengths of two sides of each triangle are equal, and the angles between the two sides (that is, the angles formed by the two sides) are equal, the two triangles are congruent triangles.

3. ASA (Angle-Side-Angle):

If two angles of each triangle are equal, and the included sides of the two angles (that is, the common side,) are equal, the two triangles are congruent triangles.

4. AAS (Angle-Angle-Side):

Two angles of each triangle are equal to each other, and the opposite sides of one of the angles (the side in the triangle except the two sides that make up the angle) or adjacent sides (that is, the side that makes up the angle) are equal to each other. If so, the two triangles are congruent triangles.

5. HL theorem (hypotenuse-leg) (hypotenuse, right-angled side):

In a right triangle, one hypotenuse and one right side are equal, and the two triangles are congruent triangles.

Reference source: Sogou Encyclopedia-Congruent Triangles

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