Proving Triangle Congruence Using the Angle-Angle-Side (AAS) Congruence Theorem

if two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent

To prove that two triangles are congruent, we use several congruence postulates and theorems

To prove that two triangles are congruent, we use several congruence postulates and theorems. One of these is the Angle-Angle-Side (AAS) congruence theorem.

The AAS congruence theorem states that if two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent.

Let’s break down the proof using the AAS congruence theorem:

Given:
Triangle ABC and triangle DEF.
∠A ≅ ∠D
∠B ≅ ∠E
Side AC ≅ Side DF

To prove: Triangle ABC ≅ triangle DEF

Proof:
1. Draw triangle ABC and triangle DEF.
(Make sure to label the vertices and sides correspondingly.)

2. Given that ∠A ≅ ∠D and ∠B ≅ ∠E, mark these angles in each triangle accordingly.

3. Since side AC ≅ side DF, mark these sides in each triangle correspondingly.

4. Now, we can see that we have two angles and a non-included side of triangle ABC congruent to the corresponding angles and side of triangle DEF.

5. According to the AAS congruence theorem, if two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent.

6. Therefore, by the AAS congruence theorem, triangle ABC ≅ triangle DEF.

With the AAS congruence theorem, we can conclude that if two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent.

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