Question

question
in the diagram below of triangle vwx, y is the midpoint of \\(\overline{vx}\\) and z is the midpoint of \\(\overline{wx}\\). if m \\(\angle xwv = 6x + 30\\), and m\\(\angle xzy = 72 - 8x\\), what is the measure of \\(\angle xwv\\)?

Explanation:

Step1: Identify Midsegment Theorem

Since \( Y \) is the midpoint of \( \overline{VX} \) and \( Z \) is the midpoint of \( \overline{WX} \), by the Midsegment Theorem, \( \overline{ZY} \parallel \overline{WV} \). Thus, \( \angle XZY \) and \( \angle XWV \) are corresponding angles, so they are equal.

$$ 6x + 30 = 72 - 8x $$

Step2: Solve for \( x \)

Add \( 8x \) to both sides:

$$ 6x + 8x + 30 = 72 $$

$$ 14x + 30 = 72 $$

Subtract 30 from both sides:

$$ 14x = 72 - 30 $$

$$ 14x = 42 $$

Divide by 14:

$$ x = \frac{42}{14} = 3 $$

Step3: Find \( m\angle XWV \)

Substitute \( x = 3 \) into \( 6x + 30 \):

$$ 6(3) + 30 = 18 + 30 = 48 $$

Answer:

The measure of \( \angle XWV \) is \( 48^\circ \).