We are asked to show the converse of Theorem 6-19. We need to show that if two of the base angles of a trapezoid are congruent, then the trapezoid is isosceles. Let's draw a trapezoid and a segment parallel to one of its legs. Let's also indicate the congruent base angles.
We will prove that ABCD is isosceles in two steps. We will show that segment AE is congruent to both AB and DC.
By construction, opposite sides of AECD are parallel, so AECD is a parallelogram. According to Theorem 6-3, this means that opposite sides are congruent.
AE≅DC
Proof of AB≅AE
Let's shift our focus to triangle △ ABE.
We will show that the three angles marked on the diagram are congruent.
Congruent Angles
Justification
∠ B≅∠ C
Given
∠ AEB≅∠ C
Segments AE and DC are parallel, and ∠ AEB and ∠ C are corresponding angles.
Since segment AE is congruent to both AB and DC, the two legs of the trapezoid are congruent, so ABCD is an isosceles trapezoid. We can summarize the process above in a flow proof.
Completed Proof
2
&Given:&& ABCD is a trapezoid withAD∥BC
& && ∠ B≅∠ C
&Prove:&& AB≅DC
Proof:
