Question

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

Explanation:

Step1: Use the angle - sum property of a triangle

The sum of interior angles of a triangle is 180°. In the left - hand triangle, we have \(x + 45+30 + 66=180\). First, find \(x\).
\[x=180-(45 + 30+66)=180 - 141=111\]

Step2: Find the angle adjacent to 119°

The angle adjacent to 119° (let's call it \(a\)) forms a linear pair with 119°. So \(a = 180 - 119=61\).

Step3: Use the angle - sum property of a triangle for the right - hand triangle

In the right - hand triangle, we know one angle is 43° and another is \(a = 61°\). Using the fact that the sum of interior angles of a triangle is 180°, we find \(y+z+43 + 61=180\). Also, we know \(z = 76\). Then \(y=180-(43 + 61+76)=180 - 180=0\) (This is wrong. Let's start over for the right - hand triangle).
In the right - hand triangle, using the angle - sum property of a triangle: \(y+z+43=180\). We know that the angle adjacent to 119° is \(180 - 119 = 61\).
The sum of angles in the right - hand triangle gives \(y+z=180 - 43=137\).
We know that the non - labeled angle in the middle triangle (where 30° and 45° are) and the angle adjacent to 119° and the angles \(y\) and \(z\) are related.
The sum of angles around the non - labeled vertex in the middle is 360°. But we can also use the triangle property directly for the right - hand triangle.
Since the sum of angles in the right - hand triangle is 180° and one angle is 43° and we assume \(z = 76\), then \(y=180-(43 + 76)=61\)

Answer:

\(x = 111\), \(y = 61\), \(z = 76\)