Question

1. in the diagram, lines $overline{pr}$ and $overline{st}$ intersect at point q such that $mangle sqp = 5(x + 8)$ and $mangle rqt=7x - 4$. (a) algebraically determine the value of x. (b) find the measure of $angle rqt$. (c) find the measure of $angle pqt$. n - gen math 8, unit 2 - tools of geometry - practice with angle pairs cmathinstruction, red hook, ny 12571, © 2022

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So, $m\angle SQP=m\angle RQT$. Set up the equation $5(x + 8)=7x-4$.

Step2: Expand the left - hand side

Using the distributive property $a(b + c)=ab+ac$, we have $5x+40 = 7x-4$.

Step3: Isolate the variable terms

Subtract $5x$ from both sides: $5x+40-5x=7x - 4-5x$, which simplifies to $40 = 2x-4$.

Step4: Solve for $x$

Add 4 to both sides: $40 + 4=2x-4 + 4$, getting $44 = 2x$. Then divide both sides by 2: $\frac{44}{2}=\frac{2x}{2}$, so $x = 22$.

Step5: Find $m\angle RQT$

Substitute $x = 22$ into the expression for $m\angle RQT$. $m\angle RQT=7x-4=7\times22-4=154 - 4=150^{\circ}$.

Step6: Find $m\angle PQT$

$\angle PQT$ and $\angle RQT$ are supplementary (a linear pair of angles), so $m\angle PQT=180^{\circ}-m\angle RQT$. Substitute $m\angle RQT = 150^{\circ}$, then $m\angle PQT=180 - 150=30^{\circ}$.

Answer:

(a) $x = 22$
(b) $m\angle RQT=150^{\circ}$
(c) $m\angle PQT=30^{\circ}$