In quadrilateral AECD opposite sides are parallel, so it is a parallelogram. According to Theorem 6-3, this means that the opposite sides are congruent.
AE≅DC
Let's now focus on triangle △ ABE.
We know that ABCD is an isosceles trapezoid, so AB is congruent to DC. Since we already know that DC is congruent to AE, this means that △ ABE is an isosceles triangle. According to the Isosceles Triangle Theorem, this means that the angles opposite to the congruent sides are congruent.
∠ B≅∠ 1
We can also compare ∠ 1 to ∠ C.
These are corresponding angles, and since AE and DC are parallel, these angles are congruent by the Corresponding Angles Theorem.
∠ 1≅∠ C
Using the Transitive Property of Congruence, we can conclude that the base angles ∠ B and ∠ C are congruent.
∠ B≅∠ C
Let's now focus on the other pair of base angles, starting with the angle ∠ D.
Let's compare this to angle ∠ C.
Since ∠ D and ∠ C are same-side interior angles and AD is parallel to BC, the Same-Side Interior Angles Postulate tells us that these are supplementary angles.
m∠ D+m∠ C=180
Similarly, angles ∠ BAD and ∠ B are also supplementary.
m∠ BAD+m∠ B=180
Since angles ∠ B and ∠ C are congruent, their supplements must also be congruent.
∠ BAD≅∠ D
The relationship between the sides and angles of the isosceles trapezoid ABCD is illustrated below.
We can summarize the process above in a flow proof.
Completed Proof
2
&Given:&& ABCD is an isosceles trapezoid
&Prove:&& ∠ B≅∠ C
& && ∠ BAD≅∠ D
Proof:
