Question

1. write the measure of the complement of each angle whose measure is given. a. 25° b. 75° c. y° 8. write the measure of the supplement of each angle whose measure is given. a. 40° b. 162° c. (50 - x)° 9. ∠a and ∠b form a linear - pair. find the measure of each angle if m∠a = w and m∠b = w - 30 (include a diagram). 10. given: mo⊥mp; lmn is a line; m∠pmn = 40° find: m∠pmo m∠omn m∠oml m∠pml 11. pr bisects ∠qps, the m∠qpr = x - 8, and the m∠qps = 5x - 100. find the value of x and m∠qpr. (include a diagram)

Explanation:

Step1: Recall angle - bisector property

If a ray bisects an angle, the measure of the two resulting angles is half of the measure of the original angle. So, $m\angle QPR=\frac{1}{2}m\angle QPS$.

Step2: Substitute the given expressions

We are given that $m\angle QPR = x - 8$ and $m\angle QPS=5x - 100$. Substituting into the equation from Step 1 gives $x - 8=\frac{1}{2}(5x - 100)$.

Step3: Solve the equation for x

Multiply both sides of the equation by 2 to get rid of the fraction: $2(x - 8)=5x - 100$. Expand the left - hand side: $2x-16 = 5x - 100$. Subtract $2x$ from both sides: $-16=3x - 100$. Add 100 to both sides: $3x=84$. Divide both sides by 3: $x = 28$.

Step4: Find the measure of $\angle QPR$

Substitute $x = 28$ into the expression for $m\angle QPR$. So, $m\angle QPR=x - 8=28 - 8=20$.

Answer:

$x = 28$, $m\angle QPR=20^{\circ}$