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Big Ideas Math Integrated I, 2016 View details

6. Proving Triangle Congruence by ASA and AAS

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Exercise 27 Page 631

According to the Converse of the Base Angles Theorem, if two angles of a triangle are congruent, then the sides opposite them are congruent.

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According to the Converse of the Base Angles Theorem, if two angles of a triangle are congruent, then the sides opposite them are congruent. Let's draw a triangle with two congruent angles.

By drawing an angle bisector, AD, we create an additional two pairs of congruent angles, ∠ BAD and ∠ CAD which we have marked below.

We now have two smaller triangles, △ ABD and △ ACD. We notice that they share a side, AD. By the Reflexive Property of Congruence, we know this is a congruent corresponding side in the triangles.

Now we have enough information to prove congruence by the AAS Congruence Theorem. Since, AB and AC are corresponding sides, we can claim that AB≅ AC which is what the Converse of the Base Angles Theorem says.

Alternative Solution

Two-Column Proof

Let's write this as a two-column proof as well.

Statement	Reason
1. Draw AD, the angle bisector of ∠ ABC	1. Construction of angle bisector
2. ∠ CAD≅ ∠ BAD	2. Definition of angle bisector
3. ∠ B≅ ∠ C	3. Given
4. AD≅ AD	4. Reflexive Property of Congruence
5. △ ABD ≅ △ ACD	5. AAS Congruence Theorem
6. AB≅ AC	6. Corresponding parts of congruent triangles are congruent

Subchapter links

Communicate Your Answer p.625

Monitoring Progress p.627-629

Exercises p.630-632

Exercise 27 Page 631

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Alternative Solution