Question

in the diagram below of triangle uvw, x is the midpoint of \\(\overline{uw}\\) and y is the midpoint of \\(\overline{vw}\\). if m \\(\angle wuv = -41 + 8x\\), and m\\(\angle wxy = -3x + 58\\), what is the measure of \\(\angle wuv\\)?

Explanation:

Step1: Identify the Midline Theorem

Since \( X \) is the midpoint of \( \overline{UW} \) and \( Y \) is the midpoint of \( \overline{VW} \), by the Midline Theorem, \( XY \parallel UV \). Therefore, \( \angle WXY \) and \( \angle WUV \) are corresponding angles, so they are equal. So we set \( -41 + 8x=-3x + 58 \).

Step2: Solve for \( x \)

Add \( 3x \) to both sides: \( -41 + 8x+3x=-3x + 58+3x \), which simplifies to \( -41 + 11x = 58 \).
Add 41 to both sides: \( -41 + 11x+41 = 58+41 \), so \( 11x=99 \).
Divide both sides by 11: \( x = \frac{99}{11}=9 \).

Step3: Find \( m\angle WUV \)

Substitute \( x = 9 \) into \( m\angle WUV=-41 + 8x \):
\( m\angle WUV=-41 + 8\times9=-41 + 72 = 31 \).

Answer:

\( 31 \)