Q:

# Complete the paragraph proof. we are given ab ≅ ae and bc ≅ de. this means abe is an isosceles triangle. base angles in an isosceles triangle are congruent based on the isosceles triangle theorem, so ∠abe ≅ ∠aeb. we can then determine △abc ≅ △aed 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, ∠acd ≅ ∠adc.

Accepted Solution

A:
Answer: Since segment AB and segment AE are congruent (given), the triangle ABE must be isoscles by definition of an isosceles triangle. From that it follows that angle ABC must be congruent to angle AED, again by definition of an isosceles triangle. Then because you are given that segment BC and segment DE are congruent, triangle ABC must be congruent to triangle AED by SAS. Now you aren't clear whether angle 1 is angle ACB or ACD. Assume it is ACB, then ACB is congruent to ADE by CPCT. Therefore angle 1 equals angle 2. QED. If angle 1 is ACD and angle 2 is ADC, then since angle ACB is supplementary to angle ACD, angle ADE is supplementary to angle ADC, and from the step above angle ACB is congruent to angle ADE, then angle ACD is congruent to angle ADC by transitive equality. Therefore angle 1 equals angle 2. QED.