Question

solve for x, y, and z.

x = type your answer...

y = type your answer...

z = type your answer...

Explanation:

Step1: Find \(x\) using triangle angle - sum property

In the left - hand triangle, the sum of angles in a triangle is \(180^{\circ}\).
We know two angles: \(31^{\circ}\) and \(61^{\circ}\). Let the third angle be \(x\).
Using the formula \(x = 180-(31 + 61)\)

$$x=180 - 92=88^{\circ}$$

Step2: Find \(y\) using triangle angle - sum property

In the right - hand triangle, we know one angle is \(25^{\circ}\) and assume the triangle is a non - overlapping part of the original figure.
Since the sum of angles in a triangle is \(180^{\circ}\), and we assume the line is a straight line (no other information to suggest otherwise for the adjacent angles \(x\) and \(y\) in terms of a linear pair, but using the triangle on the right. Wait, no, correct approach:
The two triangles (left and right) are part of a larger triangle. But using the right - hand triangle (with angle \(25^{\circ}\) and the other angle related to the larger triangle's properties. Wait, no, using the fact that the sum of angles in a triangle:
For the right - hand triangle, assume it's a triangle. The sum of angles in a triangle is \(180^{\circ}\). Let's use the property that the two small triangles (left and right) are formed by a line from the vertex.
We know that \(x + y=180\) (if they were a linear pair, but no, wait no. Wait, using the left - hand triangle: sum of angles \(31+61 + x=180\) (already done \(x = 88\)). For the right - hand triangle: sum of angles \(25+z + y=180\). Also, using the property of the larger triangle: \((31 + 25)+61+z=180\) (sum of angles in a triangle). First, find \(z\) from the larger triangle: \(31+25 + 61+z=180\), \(117+z=180\), \(z = 63\). Then for the right - hand triangle: \(y=180-(25 + z)\), \(y=180-(25 + 63)=92\)

Step3: Verify \(z\) using the larger triangle

The sum of angles in the larger triangle: \((31 + 25)+61+z=180\)

$$56+61+z=180$$

$$117+z=180$$

$$z=180 - 117=63^{\circ}$$

Answer:

\(x = 88\), \(y = 92\), \(z = 63\)