Question

triangle proofs quick check
use the image to answer the question.
given: isosceles △abc with ab = ac
prove: ∠b = ∠c
statements reasons

1. isosceles △abc with ab = ac 1. given
2. d is the mid - point of bc 2. construction
3. ad = ad 3. reflexive property of congruence
4. △abd ≅ △acd 4. sss congruence theorem
5. ∠b = ∠c 5. cpctc

Explanation:

Step1: Draw an angle - bisector

Draw angle - bisector $AD$ of $\angle BAC$. In $\triangle ABD$ and $\triangle ACD$, we know that $AB = AC$ (given), $\angle BAD=\angle CAD$ (by construction of angle - bisector), and $AD = AD$ (common side).

Step2: Apply SAS congruence

By the Side - Angle - Side (SAS) congruence criterion, $\triangle ABD\cong\triangle ACD$. That is, since we have two pairs of equal sides and the included angles equal, the two triangles are congruent.

Step3: Use congruent - triangle property

Since $\triangle ABD\cong\triangle ACD$, corresponding parts of congruent triangles are equal. So, $\angle B=\angle C$.

Answer:

The proof that $\angle B = \angle C$ in $\triangle ABC$ with $AB = AC$ is valid by constructing the angle - bisector of $\angle BAC$, showing $\triangle ABD\cong\triangle ACD$ using SAS congruence, and then using the property of corresponding parts of congruent triangles.