Question

Given:$\overline{WX}\perp \overline{UV};\overline{WX}\perp \overline{YZ};\overline{WU}\cong \overline{WV}$

Prove whatever you can about angles 1,2,3, and 4

Short answer

Statements which can be proved are$\angle 1\cong \angle 2$ and$\angle 3\cong \angle 4$

Step-by-step solution

Step 1. Show that ΔUWX≅ΔVWX.

By reflexive property of congruence,a side is congruent to itself

So, $\overline{WX}\cong \overline{WX}$

Also using given $m\angle 5=m\angle 6=90°$

Thus,$\mathrm{\Delta }UWX\cong \mathrm{\Delta }VWX$ by SAS (Side-Angle-Side) congruency criteria

Step 2. Show that ∠3≅∠4.

Since, corresponding parts of congruent triangles are congruent

So, $\angle 3\cong \angle 4$

Step 3. Show that ∠1≅∠2.

Using given statement

$m\angle WXY=m\angle WXZ=90°$

Thus, $\angle 1+\angle 3\cong \angle 2+\angle 4$

Substitute$\angle 3$ for$\angle 4$ in this equation

$\begin{array}{c}\angle 1+\angle 3\cong \angle 2+\angle 3\\ \angle 1\cong \angle 2\end{array}$

Thus, statements which can be proved are$\angle 1\cong \angle 2$ and$\angle 3\cong \angle 4$