Question

Given:$\overline{WX}\perp \overline{YZ};\angle 1\cong \angle 2;\overline{UX}\cong \overline{VX}$

Which one(s) of the following statements must be true?

(1)$\overline{XW}\perp \overline{UV}$

(2)$\overline{UV}\parallel \overline{YZ}$

(3)$\overline{VX}\perp \overline{UX}$

Short answer

True statements are (1) $\overline{XW}\perp \overline{UV}$ and (2)$\overline{UV}\parallel \overline{YZ}$

Step-by-step solution

Step 1. Show that ∠3≅∠4.

Using given statement

$m\angle WXY=m\angle WXZ=90°$

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

Substitute$\angle 1$ for$\angle 2$ in this equation

$\begin{array}{c}\angle 1+\angle 3\cong \angle 1+\angle 4\\ \angle 3\cong \angle 4\end{array}$

Step 2. Show that ∠U+∠3=90°.

In a triangle angle opposite to congruent sides are congruent.

So in $\mathrm{\Delta }UXV$, as $\overline{UX}\cong \overline{VX}$, implies $\angle U\cong \angle V$

Also, sum of angles of a triangle is $180°$

$$\begin{gathered} \angle U+\angle 3+\angle 4+\angle V=180° \\ 2\angle U+2\angle 3=180° \\ \angle U+\angle 3=90° \end{gathered}$$

Step 3. Show that WX¯⊥UV¯.

Since, sum of angles of a triangle is $180°$

In$\mathrm{\Delta }UXW$

$$\begin{gathered} \angle U+\angle 3+\angle UWX=180° \\ 90°+\angle 5=180° \\ \angle 5=90° \end{gathered}$$

Thus,$\overline{XW}\perp \overline{UV}$

Step 4. Show that UV¯∥YZ¯.

Since, $\overline{XW}$is perpendicular to both $\overline{YZ}\mathrm{and}\overline{UV}$

Thus,$\overline{UV}\parallel \overline{YZ}$