When two angles and corresponding side of triangle is equal then it is said to be congruent by AAS CONGRUENCE

Theorem 7.1:

The ASA (Angle-Side-Angle) Congruence Rule states that if two angles and the included side of one triangle are congruent to the corresponding angles and the included side of another triangle, then the two triangles are congruent.

Explanation:

To understand the ASA Congruence Rule, let's consider an example. Suppose we have two triangles, ΔABC and ΔDEF, where ∠A = ∠D, ∠B = ∠E, and side AB = side DE.

Proof:

Step 1: Given that ∠A = ∠D and ∠B = ∠E. This means that the two pairs of corresponding angles are congruent.

Step 2: Given that AB = DE. This means that the included side between the congruent angles is equal in length.

Step 3: By the ASA Congruence Rule, if two angles and the included side of one triangle are congruent to the corresponding angles and the included side of another triangle, then the two triangles are congruent.

Therefore, by applying the ASA Congruence Rule, we can conclude that ΔABC is congruent to ΔDEF.

Example:

Let's consider an example to further understand the ASA Congruence Rule. Suppose we have two triangles, ΔPQR and ΔXYZ, where ∠P = ∠X, ∠Q = ∠Y, and side PQ = side XY.

Solution:

By applying the ASA Congruence Rule, we can conclude that ΔPQR is congruent to ΔXYZ. This means that all corresponding sides and angles of the two triangles are congruent.

Key Points:

- ASA Congruence Rule states that if two angles and the included side of one triangle are congruent to the corresponding angles and the included side of another triangle, then the two triangles are congruent.
- The ASA Congruence Rule is one of the methods used to prove the congruence of triangles.
- In the ASA Congruence Rule, the included side is the side between the two given congruent angles.
- The ASA Congruence Rule is based on the fact that if two triangles have congruent corresponding parts, then the triangles themselves are congruent.