Question

1. isosceles \\( \triangle a b c \\) with \\( a b \cong a c \\) 1. given
2. \\( d \\), the midpoint of \\( \overline{b c} \\) 2. construction

\\( \overline{a d} \\) through points \\( a \\) and \\( d \\)

1. \\( \overline{b d} \cong \overline{d c} \\) 3. definition of midpoint
2. \\( \overline{a d} \cong \overline{a d} \\) 4. reflexive property of congruence
3. \\( \triangle a b d \cong \triangle a c d \\) 5. sas congruence theorem
4. \\( \angle b \cong \angle c \\) 6. cpctc theorem

(1 point)
reason 5 has a mistake. it should say \sss congruence theorem.\
statement 3 has a mistake. it should say \\\( \angle b a d \cong \angle c a d \\).\
statement 6 has a mistake. it should say \\\( \angle a \cong \angle c \\).\

Explanation:

Step1: Analyze the given information

We have an isosceles triangle \(ABC\) with \(AB\cong AC\), \(D\) is the mid - point of \(BC\) (so \(BD\cong DC\)), and \(AD\) is a common side (\(AD\cong AD\)).

Step2: Recall congruence theorems

• SAS (Side - Angle - Side): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
• SSS (Side - Side - Side): If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

In \(\triangle ABD\) and \(\triangle ACD\), we have \(AB = AC\) (given), \(BD=DC\) (by the definition of mid - point), and \(AD = AD\) (reflexive property). So, we are using the SSS (Side - Side - Side) congruence theorem, not the SAS (Side - Angle - Side) congruence theorem.

Answer:

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