Question

find the measure of ∠yvx.

Explanation:

Step1: Note that a straight - line angle is 180°.

The angle $\angle UVW = 180^{\circ}$.

Step2: Express $\angle UVW$ in terms of smaller angles.

$\angle UVW=\angle UVX+\angle XVY+\angle YVZ+\angle ZVW$. We know that $\angle UVX = 49^{\circ}$, $\angle YVZ=52^{\circ}$, $\angle ZVW = 59^{\circ}$. Let $\angle XVY=x$. Then $180^{\circ}=49^{\circ}+x + 52^{\circ}+59^{\circ}$.

Step3: Solve for $x$.

First, add the known angles on the right - hand side: $49^{\circ}+52^{\circ}+59^{\circ}=160^{\circ}$. Then $x=180^{\circ}-160^{\circ}=20^{\circ}$. But we made a wrong start. Since $\angle UVY = 90^{\circ}$ (it is a right - angle as indicated by the symbol), and $\angle UVX = 49^{\circ}$.
We know that $\angle YVX=\angle UVY-\angle UVX$.

Step4: Calculate $\angle YVX$.

$\angle YVX = 90^{\circ}-49^{\circ}=39^{\circ}$.

Answer:

$39^{\circ}$