Question

question
in the diagram below of triangle pqr, s is the midpoint of \\(\overline{pr}\\) and t is the midpoint of \\(\overline{qr}\\). if m\\(\angle rpq = 34 - 2x\\), and m\\(\angle rst = -5x + 46\\), what is the measure of \\(\angle rpq\\)?
answer attempt 1 out of 2
m\\(\angle rpq = \square\\)\\(^\circ\\)
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Explanation:

Step1: Identify the relationship (Midline Theorem)

Since \( S \) is the midpoint of \( \overline{PR} \) and \( T \) is the midpoint of \( \overline{QR} \), by the Midline Theorem, \( \overline{ST} \parallel \overline{PQ} \). Therefore, \( \angle RPQ \) and \( \angle RST \) are corresponding angles, so they are equal. Thus, we set up the equation:
\( 34 - 2x = -5x + 46 \)

Step2: Solve for \( x \)

Add \( 5x \) to both sides:
\( 34 - 2x + 5x = 46 \)
Simplify:
\( 34 + 3x = 46 \)
Subtract 34 from both sides:
\( 3x = 46 - 34 \)
\( 3x = 12 \)
Divide both sides by 3:
\( x = \frac{12}{3} = 4 \)

Step3: Find \( m\angle RPQ \)

Substitute \( x = 4 \) into the expression for \( m\angle RPQ \):
\( m\angle RPQ = 34 - 2x \)
\( m\angle RPQ = 34 - 2(4) \)
\( m\angle RPQ = 34 - 8 \)
\( m\angle RPQ = 26 \)

Answer:

\( 26 \)