Question

use the image to answer the question.
for is working to prove the base angles theorem. his proof is shown below. critique his reasoning. which statement or reason in his proof has a mistake? how can he fix his mistake?
given: isosceles \\( \triangle a b c \\) with \\( \overline{a b} \cong \overline{a c} \\)
prove: \\( \angle b \cong \angle c \\)
identity mistakes in steps to prove congruency between angles
\\( \square \\) (1 point)
\\( \square \\) statement 6 has a mistake. it should say \\( \angle a \cong \angle c \\)
\\( \square \\) reason 3 has a mistake. it should say definition of angle bisector
\\( \square \\) statement 2 has a mistake. it should say \\( \angle b a d \cong \angle c a d \\)
\\( \square \\) reason 5 has a mistake. it should say sss congruence theorem

Explanation:

Step1: Analyze the given information

We are given \(\triangle ABC\) with \(\overline{AB}\cong\overline{AC}\), \(D\) is the mid - point of \(\overline{BC}\) (so \(\overline{BD}\cong\overline{DC}\)), and \(\overline{AD}\cong\overline{AD}\) (reflexive property).

Step2: Recall congruence theorems

The SSS (Side - Side - Side) congruence theorem states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. The SAS (Side - Angle - Side) congruence theorem requires two sides and the included angle to be congruent.
In this case, we have \(\overline{AB}\cong\overline{AC}\), \(\overline{BD}\cong\overline{DC}\), and \(\overline{AD}\cong\overline{AD}\). There is no information about an included angle. So, we should use the SSS congruence theorem instead of the SAS congruence theorem for \(\triangle ABD\cong\triangle ACD\).

Answer:

D. Reason 5 has a mistake. It should say "SSS Congruence Theorem"