Question

given: \\( \overline { a b } \cong \overline { a e } ; \overline { b c } \cong \overline { 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 \\( \overline { b c } \cong \overline { 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. because of cpctc, segment ac is congruent to segment. 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 of triangles

We know \(AB\cong AE\), \(BC\cong DE\), and \(\angle ABE\cong\angle AEB\). By the Side - Angle - Side (SAS) congruence criterion (\(AB = AE\), \(\angle ABC=\angle AED\), \(BC = DE\)), \(\triangle ABC\cong\triangle AED\)

Step2: Use CPCTC

Since \(\triangle ABC\cong\triangle AED\) by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \(AC\cong AD\)

Answer:

The first blank is filled with \(SAS\) (Side - Angle - Side), and the second blank is filled with \(AD\)