Question

$\overrightarrow{qs}$ bisects $\angle pqr$. use the diagram and the given angle measure to find the indicated angle measures. (see example 5.) 34) $m\angle rqs = 71^\circ$. find $m\angle pqs$ and $m\angle pqr$. 35) $m\angle pqr = 124^\circ$. find $m\angle pqs$ and $m\angle rqs$.

Explanation:

Step1: Recall angle bisector property

A bisector splits an angle into two congruent angles, so $m\angle PQS = m\angle RQS$, and $m\angle PQR = m\angle PQS + m\angle RQS$.

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For question 34:

Step1: Set equal bisected angles

Since $\overline{QS}$ bisects $\angle PQR$, $m\angle PQS = m\angle RQS$.
$m\angle PQS = 71^\circ$

Step2: Calculate total angle

Sum the two congruent angles.
$m\angle PQR = 71^\circ + 71^\circ = 142^\circ$

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For question 35:

Step1: Split total angle equally

The bisector divides $\angle PQR$ into two equal parts.
$m\angle PQS = m\angle RQS = \frac{m\angle PQR}{2}$

Step2: Compute each angle

Substitute $m\angle PQR = 124^\circ$.
$m\angle PQS = m\angle RQS = \frac{124^\circ}{2} = 62^\circ$

Answer:

1. $m\angle PQS = 71^\circ$, $m\angle PQR = 142^\circ$
2. $m\angle PQS = 62^\circ$, $m\angle RQS = 62^\circ$