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 angle - sum property of a triangle

The sum of interior angles of a triangle is 180°. In the left - hand triangle, we have \(x+45 + 24=180\) (since the angle adjacent to 66° in the left - hand triangle is \(180 - 66=114\) and the third angle of the left - hand triangle is \(180-(114 + 30)=36\), and \(x = 180-(45 + 24)=111\)).

Step2: Consider the middle triangle

In the middle triangle, we know one angle is 30° and another is 43°. Let's find the third angle of this triangle. The sum of interior angles of a triangle is 180°. So the third angle of the middle triangle is \(180-(30 + 43)=107\).

Step3: Use linear pair and angle - sum property

The angle adjacent to \(y\) in the right - hand triangle and the 107° angle from the middle triangle form a linear pair. So the angle adjacent to \(y\) is \(180 - 107 = 73\). In the right - hand triangle, using the angle - sum property of a triangle (\(180\)°), we have \(y+73+119 = 180\), so \(y=180-(73 + 119)=180 - 192=- 12\) (this is wrong. Let's use another approach).
We know that the sum of angles around a point is 360°. Consider the point where the three triangles meet. The angles around this point are part of the interior angles of the triangles.
The sum of angles in the three - triangle intersection:
The non - \(x,y,z\) angles around the intersection point sum up to \(45+30 + 43=118\).
We know that the sum of angles around a point is 360°.
For \(y\):
We consider the angles in the right - hand triangle. The sum of interior angles of a triangle is 180°. The angle adjacent to the 119° angle in the right - hand triangle is \(180 - 119 = 61\).
In the right - hand triangle, using the angle - sum property \(y+43+61=180\), so \(y = 76\).

Answer:

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