Question

Are the triangles below congruent? explain what theorem proves the triangles are congruent, or why the triangles are not congruent.

Answer 1

In the given figure, it appears that the sides OP and RQ are congruent and parallel. The two sides are being intersected by the diagonal OQ.

Extending sides OP, RQ, and diagonal OQ. It is similar to parallel lines being cut by a traversal.

Similarity No. 1: Similar Angles

Looking at that perspective, we can say that ∠1 and ∠2 are pairs of alternate angles and are congruent.

`\angle1\text{ }\cong\text{ }\angle2`

The same with ∠3 and ∠4.

`\angle3\text{ }\cong\text{ }\angle4`

Similarity No. 2: Similar Sides

It is given in the figure that the two triangles share one side which is OQ. Thus, it is one concrete proof that one of the sides of the triangles is equal.

Conclusion: The Triangles are therefore congruent under ASA Theorem. The two angles and their included side is similar.