Question

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

Explanation:

Step1: Use angle - sum property of a triangle

The sum of interior angles of a triangle is 180°. In the left - hand triangle, we have \(x + 45+66 = 180\).
So, \(x=180-(45 + 66)=180 - 111=69\).

Step2: Find an intermediate angle

In the middle part, consider the non - labeled angle adjacent to the 30° angle in the middle triangle. Let's call it \(a\). The sum of angles around a point is 360°. In the intersection of the three triangles, we know some angles. The non - labeled angle \(a=360-(66 + 30+43)=360 - 139 = 221\). But we are interested in the interior angle of the triangle it belongs to, which is \(180 - a\). The interior angle is \(180-(360-(66 + 30+43))=139 - 180=- 41\) (wrong approach). Let's use another way.

We know that in the right - hand triangle, we first find the third angle of the triangle with 43° and the angle adjacent to 119°. The angle adjacent to 119° is \(180 - 119=61\).

Step3: Calculate \(y\) and \(z\)

In the right - hand triangle, using the angle - sum property of a triangle (\(180^{\circ}\)), we have \(y+43+(180 - 119)=180\).
\(y+43 + 61=180\), so \(y=180-(43 + 61)=76\).
Also, since the sum of angles in the right - hand triangle is 180°, and we know two angles are 43° and 61°, then \(z = 180-(43+61)=76\).

Answer:

\(x = 69\), \(y = 76\), \(z = 76\)