Question

1. set up an equation and solve for the value of a. 10. set up an equation and solve for the value of z. 11. use the given picture to help you set up an equation and solve for x. op bisects ∠hot. m∠hop = 3x - 1 and m∠hot = 46°. 12. two angles are supplementary. one angle is 12° larger than the other one. what are the measures of the two angles? x = m∠pot =

Explanation:

9.

Step1: Set up equation

Since vertical - angles are equal, we have $5a=17a - 18$.

Step2: Rearrange terms

Subtract $5a$ from both sides: $0 = 17a-5a - 18$, which simplifies to $0 = 12a - 18$.

Step3: Solve for a

Add 18 to both sides: $18 = 12a$. Then divide both sides by 12: $a=\frac{18}{12}=\frac{3}{2}=1.5$.

Step1: Set up equation

Since vertical - angles are equal, we have $9z - 5=8z + 6$.

Step2: Solve for z

Subtract $8z$ from both sides: $9z-8z-5=8z - 8z+6$, which gives $z-5 = 6$. Then add 5 to both sides: $z=6 + 5=11$.

Step1: Use angle - bisector property

If $\overrightarrow{OP}$ bisects $\angle HOT$, then $m\angle HOP=m\angle POT$ and $m\angle HOP=\frac{1}{2}m\angle HOT$. Given $m\angle HOT = 46^{\circ}$, so $m\angle HOP=\frac{46}{2}=23^{\circ}$. Also, $m\angle HOP = 3x - 1$.

Step2: Set up equation

Set $3x-1 = 23$.

Step3: Solve for x

Add 1 to both sides: $3x=23 + 1=24$. Then divide both sides by 3: $x = 8$.

Answer:

$a = 1.5$

10.