Question

1. given $overline{mn}paralleloverline{pq}$, $mangle pts=(8x + 20)^{circ}$, and $mangle nst=(11x - 31)^{circ}$. a. relationship: ________ b. $x =$ _______ (show work) c. find the $mangle pts$. d. find the $mangle stq$.

Explanation:

Step1: Identify angle - relationship

Since $\overline{MN}\parallel\overline{PQ}$, $\angle PTS$ and $\angle NST$ are alternate - interior angles. Alternate - interior angles are congruent when two parallel lines are cut by a transversal. So the relationship is $\angle PTS=\angle NST$.

Step2: Solve for $x$

Set up the equation based on the angle - relationship:
$8x + 20=11x-31$
Subtract $8x$ from both sides:
$20 = 11x-8x-31$
$20 = 3x-31$
Add 31 to both sides:
$20 + 31=3x$
$51 = 3x$
Divide both sides by 3:
$x=\frac{51}{3}=17$

Step3: Find $m\angle PTS$

Substitute $x = 17$ into the expression for $m\angle PTS$:
$m\angle PTS=8x + 20=8\times17+20=136 + 20=156^{\circ}$

Step4: Find $m\angle STQ$

Since $\angle PTS$ and $\angle STQ$ are a linear pair, and the sum of angles in a linear pair is $180^{\circ}$.
$m\angle STQ=180 - m\angle PTS$
$m\angle STQ=180-156 = 24^{\circ}$

Answer:

a. $\angle PTS=\angle NST$
b. $x = 17$
c. $m\angle PTS = 156^{\circ}$
d. $m\angle STQ = 24^{\circ}$