Question

given: ( overline{ab} cong overline{ae}; overline{bc} cong overline{de} ) prove: ( angle acd cong angle adc ) complete the paragraph proof. we are given ( overline{ab} cong overline{ae} ) and ( overline{bc} cong overline{de} ). this means abe is an isosceles triangle. base angles in an isosceles triangle are congruent based on the isosceles triangle theorem, so ( angle abe cong angle aeb ). we can then determine ( \triangle abc cong \triangle aed ) by. because of cpctc, segment ac is congruent to segmen. triangle acd is an isosceles triangle based on the definitioeles triangle. therefore, based on the isosceles triangle the ( angle acd cong angle adc ).

Explanation:

Step1: Identify the congruent triangles

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

Step2: Use CPCTC

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

Answer:

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