Question

if \overline{ps} || \overline{qr}, find the measure of \angle rsp.
a) 54°
b) 49°
c) 59°
d) 72°
e) 24°

Explanation:

Step1: Use the property of parallel lines

Since \( \overline{PS}\parallel\overline{QR}\), then \( \angle QRP=\angle RPS\) (alternate - interior angles). So \(5y + 14=8y-13\).
Solve the equation \(5y + 14=8y-13\):
Subtract \(5y\) from both sides: \(14 = 3y-13\).
Add \(13\) to both sides: \(3y=14 + 13=27\).
Divide both sides by \(3\): \(y = 9\).

Step2: Use the triangle - angle sum property

In \(\triangle RSP\), by the angle - sum property of a triangle (\(\angle RSP+\angle SRP+\angle RPS = 180^{\circ}\)), and also \(3x+2x + 1+8y-13=180\).
Substitute \(y = 9\) into \(8y-13\): \(8\times9-13=72 - 13=59\).
Then the equation for \(x\) is \(3x+2x + 1+59=180\).
Combine like terms: \(5x+60 = 180\).
Subtract \(60\) from both sides: \(5x=120\).
Divide both sides by \(5\): \(x = 24\).

Step3: Calculate \(\angle RSP\)

Substitute \(x = 24\) into \(2x + 1\).
\(\angle RSP=2x + 1=2\times24+1=49\).

Answer:

B. \(49^{\circ}\)