Question

2 find m∠rqt. (4x + 15)° (10x - 3)° a) 59° b) 61° c) 63° d) 117° e) 119°

Explanation:

Answer:

First, since vertical - angles are congruent, we set up the equation \(4x + 15=10x - 3\).
Solve for \(x\):
\[

$$\begin{align*} 15 + 3&=10x-4x\\ 18&=6x\\ x& = 3 \end{align*}$$

\]
Then find the measure of the angle. Substitute \(x = 3\) into \(4x + 15\):
\(m\angle RQT=4x + 15=4\times3+15=12 + 15=27\) (This is wrong. We should use the linear - pair relationship. Let's start over.)

Since \(\angle PQS\) and \(\angle RQT\) are vertical angles, they are congruent. Also, \(\angle PQS\) and \((10x - 3)^{\circ}\) are vertical angles.

We know that \((4x + 15)+(10x - 3)=180\) (because they are a linear - pair of angles)
\[

$$\begin{align*} 4x+15 + 10x-3&=180\\ 14x+12&=180\\ 14x&=180 - 12\\ 14x&=168\\ x&=12 \end{align*}$$

\]
Substitute \(x = 12\) into \(4x + 15\):
\(m\angle RQT=4x + 15=4\times12+15=48 + 15=63^{\circ}\)

So the answer is C. \(63^{\circ}\)