Question

8
if $overline{ps}||overline{qr}$, find
the measure of $angle rsp$.
a) $54^circ$
pajamas
b) $49^circ$
bicycle helmets
c) $59^circ$
rain boots
d) $72^circ$
space suits
e) $24^circ$
crowns

Explanation:

Step1: Set parallel angle equality

Since $\overline{PS} \parallel \overline{QR}$, alternate interior angles are equal:
$5y + 14 = 8y - 13$

Step2: Solve for $y$

Rearrange to isolate $y$:
$14 + 13 = 8y - 5y$
$27 = 3y$
$y = \frac{27}{3} = 9$

Step3: Set triangle angle sum to $180^\circ$

In $\triangle RPS$, angles sum to $180^\circ$:
$3x + (2x + 1) + (8y - 13) = 180$

Step4: Substitute $y=9$ and solve for $x$

Substitute $y=9$:
$3x + 2x + 1 + (8(9) - 13) = 180$
$5x + 1 + 59 = 180$
$5x + 60 = 180$
$5x = 120$
$x = 24$

Step5: Calculate $\angle RSP$

$\angle RSP = 2x + 1$, substitute $x=24$:
$\angle RSP = 2(24) + 1$

Answer:

C) $59^\circ$ rain boots