Question

given \\( \overline { a b } \cong \overline { a e } \\), \\( b c \cong d e \\) prove \\( \angle a c d \cong \angle a d c \\) complete the paragraph proof we are given \\( \overline { a b } \cong \overline { a e } \\) and \\( b c \cong d e \\) this means abe is an isosceles triangle base angles in an isosceles triangle are congruent based on the isosceles triangle theorem, so \\( \angle a b e \cong \angle a e b \\) we can then determine \\( \triangle a b c \cong \triangle a e d \\) by \\( \square \\) because of cpctc, segment ac is congruent to segment \\( \square \\) triangle acd is an isosceles triangle based on the definition of isosceles triangle therefore, based on the isosceles triangle theorem, \\( \angle a c d \cong \angle a d c \\)

Explanation:

Step1: Determine the congruence criterion for \(\triangle ABC\) and \(\triangle AED\)

We know that \(AB = AE\) (given), \(\angle ABC=\angle AED\) (base angles of isosceles \(\triangle ABE\)), and \(BC = DE\) (given). By the \(SAS\) (Side - Angle - Side) congruence criterion, \(\triangle ABC\cong\triangle AED\).

Step2: Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle ABC\cong\triangle AED\) by \(SAS\), then by CPCTC, \(AC = AD\).

Answer:

1. \(SAS\) (Side - Angle - Side)
2. \(AD\)