Question

find the value of each variable.

Explanation:

Step1: Recall angle - sum properties of triangles

The sum of interior angles of a triangle is 180°. For exterior - angle property, an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.

Step2: Solve for angles in problem 7

In the first sub - triangle of problem 7, if one angle is 64° and another is 28°, then the third angle \(y\) is \(y = 180-(64 + 28)=88^{\circ}\). The exterior angle \(x\) of the larger triangle is \(x=64 + 28=92^{\circ}\). And \(z = 180 - 39=141^{\circ}\).

Step3: Solve for angles in problem 8

Let's consider the intersection of the two triangles in problem 8. The vertical angles are equal. Let the unlabeled angle at the intersection be \(a\). The angle adjacent to 150° is \(180 - 150=30^{\circ}\). In the left - hand triangle, if one angle is 55° and the other is 30°, then \(x=180-(55 + 30)=95^{\circ}\).

Step4: Solve for angles in problem 9

In problem 9, the exterior angle is 34°. Let the non - right interior angle of the right - triangle be \(y\). Then \(y = 34^{\circ}\) (alternate interior angles or exterior angle property). And \(x=90 - 34 = 56^{\circ}\).

Step5: Solve for angles in problem 10

In the right - triangle part of problem 10, if one non - right angle is \(x\) and the other is 65°, then \(x = 90-65 = 25^{\circ}\). In the larger triangle, if one angle is 36° and another is 90°, then the third angle \(y=180-(36 + 90)=54^{\circ}\).

Answer:

For problem 7: \(x = 92^{\circ}\), \(y = 88^{\circ}\), \(z = 141^{\circ}\)
For problem 8: \(x = 95^{\circ}\)
For problem 9: \(x = 56^{\circ}\), \(y = 34^{\circ}\)
For problem 10: \(x = 25^{\circ}\), \(y = 54^{\circ}\)