Question

4 - use the diagram to answer the questions.

1. name an angle supplementary to \\( \angle t v y \\).
2. name a pair of vertical angles with vertex \\( w \\).
3. name a pair of adjacent angles with vertex \\( v \\).
4. if \\( m \angle s x w = 5 x - 16 \\), find the value of \\( x \\) so that \\( \overline { s x } \perp \overline { w y } \\).

Explanation:

Step1: Find supplementary angle to ∠TVY

Supplementary angles sum to \(180^{\circ}\). ∠RVY shares a side with ∠TVY and forms a linear pair (sum \(180^{\circ}\)).

Step2: Find vertical angles at vertex W

Vertical angles are opposite angles formed by two intersecting lines. ∠QWP and ∠YWQ are vertical angles at vertex W.

Step3: Find adjacent angles at vertex V

Adjacent angles share a common vertex and side. ∠RVY and ∠YVT share vertex V and side VY.

Step4: Solve for x when \( \overline{SX}\perp\overline{WY}\)

If \( \overline{SX}\perp\overline{WY}\), then \(m\angle SXW = 90^{\circ}\). Set \(5x - 16=90\).

$$\begin{align*} 5x-16&=90\\ 5x&=90 + 16\\ 5x&=106\\ x&=\frac{106}{5}=21.2 \end{align*}$$

Answer:

1. \( \angle RVY\)
2. \( \angle QWP\) and \( \angle YWQ\)
3. \( \angle RVY\) and \( \angle YVT\) (answers may vary)
4. \(x = 21.2\)