Question

for questions #7 - 9, refer to the diagram at the right, where $overline{pq}perpoverline{qr}$, $mangle pqt = 2x - 5$, $mangle tqs = 4x + 5$, and $mangle sqr = 3x$. 7. find $mangle pqr$. 8. find $mangle tqr$. 9. find the complement of $angle pqs$ in degrees.

Explanation:

Step1: Since $\overline{PQ}\perp\overline{QR}$, $m\angle PQR = 90^{\circ}$

This is by the definition of perpendicular lines forming a right - angle.

Step2: Find the value of $x$

We know that $m\angle PQT+m\angle TQS+m\angle SQR=m\angle PQR$. Substituting the given angle measures: $(2x - 5)+(4x + 5)+3x=90$.
Combining like terms: $2x-5 + 4x+5+3x=90$, which simplifies to $9x=90$.
Dividing both sides by 9 gives $x = 10$.

Step3: Find $m\angle TQR$

$m\angle TQR=m\angle TQS+m\angle SQR$. Substitute $x = 10$ into the expressions for $m\angle TQS$ and $m\angle SQR$.
$m\angle TQS=4x + 5=4\times10+5=45^{\circ}$ and $m\angle SQR=3x=3\times10 = 30^{\circ}$.
So $m\angle TQR=45^{\circ}+30^{\circ}=75^{\circ}$.

Step4: Find $m\angle PQS$

$m\angle PQS=m\angle PQT+m\angle TQS$. Substitute $x = 10$ into the expressions for $m\angle PQT$ and $m\angle TQS$.
$m\angle PQT=2x - 5=2\times10-5 = 15^{\circ}$ and $m\angle TQS=45^{\circ}$, so $m\angle PQS=15^{\circ}+45^{\circ}=60^{\circ}$.
The complement of an angle $\theta$ is $90^{\circ}-\theta$.
The complement of $\angle PQS$ is $90 - 60=30^{\circ}$.

Answer:

1. $90^{\circ}$
2. $75^{\circ}$
3. $30^{\circ}$