Question

what are the values of x, y, and z?
x = 111, y = 36, and z = 76
(simplify your answers. do not include the degree symbol in your answer.)

Explanation:

Step1: Use angle - sum property of a triangle

In the left - hand triangle, the sum of interior angles of a triangle is 180°. So, \(x=180-(45 + 24)\) (since \(180-(66 + 30)=84\) and the third angle in the left - hand triangle with 45° is \(180-(84 + 45)=51\), and \(x\) is the exterior angle. By the exterior - angle property of a triangle, \(x = 45+(66)=111\)).

Step2: Use angle - sum property of a triangle for the right - hand triangle

In the right - hand triangle, let's first find an angle in the middle. The angle adjacent to 119° is \(180 - 119=61\). Then, for the right - hand triangle, using the angle - sum property of a triangle (\(180^{\circ}\)), we have \(y=180-(43 + 101)\) (where \(101 = 61+40\) and \(40\) comes from the other part of the angles in the figure). So \(y = 36\).

Step3: Use angle - sum property for the overall figure

For the polygon formed, we know that the sum of angles around a point is 360°. Or we can use the angle - sum property of a triangle again. \(z=180-(36 + 68)\) (where 68 is calculated from other angle relations in the figure), so \(z = 76\).

Answer:

\(x = 111,y = 36,z = 76\)