We can see that triangles △ Z W M and △ Y X M have two pairs of congruent sides, and the included angles are also congruent. According to the Side-Angle-Side (SAS) Congruence Postulate, this means that the two triangles are congruent.
△ Z W M≅△ Y X M
We know that corresponding sides of congruent triangles are congruent.
Z M≅Y M
Since these are two sides of triangle △ Z M Y, this proves that △ Z M Y is isosceles. We can summarize the steps above in a paragraph proof.
Completed Proof
2
&Given:&& WXYZis a parallelogram
& && ∠ W≅∠ X
& && Mis the midpoint ofWX
&Prove:&& △ ZMYis isosceles
Proof.
Opposite sides of parallelogram WXYZ are congruent, so ZW≅YX. It is given that M is the midpoint of segment WX, so MW≅MX. It is also given that ∠ W≅∠ X, so according to the Side-Angle-Side (SAS) Congruence Postulate, △ ZWM≅△ YXM. Corresponding sides of congruent triangles are congruent, so ZM≅YM. These are two sides of triangle △ ZMY, so this triangle is isosceles.
