Question

1. (mangle xyz = 117^{circ}). find (mangle xyw) and (mangle wyz)

Explanation:

Step1: Set up equation based on angle - addition postulate

Since $\angle XYZ=\angle XYW+\angle WYZ$, we have $(6x + 44)+(-10x + 65)=117$.

Step2: Combine like - terms

$6x-10x+44 + 65=117$, which simplifies to $-4x+109 = 117$.

Step3: Solve for $x$

Subtract 109 from both sides: $-4x=117 - 109$, so $-4x=8$. Then divide both sides by $-4$, we get $x=-2$.

Step4: Find $m\angle XYW$

Substitute $x = - 2$ into the expression for $\angle XYW$: $m\angle XYW=6x + 44=6\times(-2)+44=-12 + 44 = 32^{\circ}$.

Step5: Find $m\angle WYZ$

Substitute $x=-2$ into the expression for $\angle WYZ$: $m\angle WYZ=-10x + 65=-10\times(-2)+65=20 + 65 = 85^{\circ}$.

Answer:

$m\angle XYW = 32^{\circ}$, $m\angle WYZ = 85^{\circ}$