Question

find the values of the missing angles in the diagram below.

Explanation:

Step1: Use vertical - angles property

Vertical angles are equal. The angle opposite to the $77^{\circ}$ angle is also $77^{\circ}$. And the angle $x$ and the angle composed of $32^{\circ}$ and the vertical - angle of $77^{\circ}$ are on a straight - line. A straight - line has an angle of $180^{\circ}$. So $x=180-(32 + 77)=71^{\circ}$.

Step2: Use corresponding - angles property

Assume the two parallel lines (indicated by the arrowheads). The angle corresponding to the $98^{\circ}$ angle (not shown in full working for simplicity of diagram) and the angle $y$ are related. Since the sum of angles on a straight - line is $180^{\circ}$, and the angle corresponding to $98^{\circ}$ and $y$ are supplementary. So $y = 98^{\circ}$ (corresponding angles for parallel lines).

Step3: Use triangle - angle - sum property

In the triangle formed by the intersection of the lines, we know one angle is $32^{\circ}$ (from the given), another angle is $y = 98^{\circ}$, and let the third angle be $z$. The sum of angles in a triangle is $180^{\circ}$. So $z=180-(98 + 32)=50^{\circ}$.

Answer:

$x = 71^{\circ}$, $y = 98^{\circ}$, $z = 50^{\circ}$