Question

1. (overline{qs}) bisects (angle pqr). if (mangle pqs = 5x) and (mangle rqs=2x + 6), then what is (mangle pqr)?

Explanation:

Step1: Use the angle - bisector property

Since \(\overrightarrow{QS}\) bisects \(\angle PQR\), then \(m\angle PQS=m\angle RQS\).
So, \(5x = 2x+6\).

Step2: Solve the equation for \(x\)

Subtract \(2x\) from both sides of the equation \(5x = 2x + 6\).
\(5x-2x=2x + 6-2x\), which gives \(3x=6\).
Divide both sides by \(3\): \(x=\frac{6}{3}=2\).

Step3: Find \(m\angle PQS\) and \(m\angle RQS\)

Substitute \(x = 2\) into \(m\angle PQS=5x\), we get \(m\angle PQS=5\times2 = 10^{\circ}\).
Substitute \(x = 2\) into \(m\angle RQS=2x + 6\), we get \(m\angle RQS=2\times2+6=4 + 6=10^{\circ}\).

Step4: Calculate \(m\angle PQR\)

Since \(m\angle PQR=m\angle PQS+m\angle RQS\) (angle - addition postulate).
\(m\angle PQR=10^{\circ}+10^{\circ}=20^{\circ}\).

Answer:

\(20^{\circ}\)