We will show the claim in two steps. First, we use the given angle congruence to show that WXYV is a trapezoid, then we us the given segment congruence to show that it is isosceles.
WXYV Is a Trapezoid
Angles ∠ W and ∠ ZXY are corresponding angles formed by transversal WZ and segments WV and XY.
It is given that angles ∠ W and ∠ ZXY are congruent, so according to the Converse of the Corresponding Angles Theorem, segments WV and XY are parallel. Since the other two sides are not parallel, by definition, quadrilateral WXYV is a trapezoid.
WXYV Is Isosceles
Let's focus now on triangle △ WZV.
It is given that sides WZ and ZV are congruent, so according to the Isosceles Triangle Theorem, angles ∠ W and ∠ V are congruent. These angles are base angles of trapezoid WXYV.
Trapezoid WXYV has two congruent base angles, so according to Theorem 6.22, it is isosceles. We can summarize the steps above in a two-column proof.
Completed Proof
2
&Given:&& WZ∥ZV
& && ∠ W≅∠ ZXY
&Prove:&& WXYV is an isosceles trapezoid
Proof:
Statements
Reasons
1.
∠ W≅∠ ZXY
1.
Given
2.
WV∥XY
2.
Converse of the Corresponding Angles Theorem
3.
WXYV is a trapezoid.
3.
Definition
4.
WZ≅ZV
4.
Given
5.
∠ W≅∠ V
5.
Isosceles Triangle Theorem
6.
WXYV is a trapezoid.
6.
Trapezoid with congruent base angles (Theorem 6.22)
Extra
We did not use all given information.
Notice that in the proof above we did not use the information that XY bisects WZ and ZV. This is not needed. If triangle △ WZX is isosceles, then no matter where we draw XY, as long as it is parallel to the base of the triangle, WXYV will be an isosceles trapezoid.
On the other hand, if we do use the information that XY bisects WZ and ZV, then the given angle congruence is not needed. If X is the midpoint of WZ and Y is the midpoint of ZV, then XY is parallel to the base of the triangle. This can be shown using similarity of triangles.
