Question

for exercises 7 - 12, find the value of x for each figure. for exercises 13 - 15. find the measure of the given angle.

Explanation:

Step1: Recall the exterior - angle property of a triangle

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.

Exercise 7:

The exterior angle \(x\) is the sum of the two non - adjacent interior angles. So \(x=70 + 38\).

$$x=108$$

Exercise 8:

The exterior angle \(x\) is the sum of the two non - adjacent interior angles. So \(x=39+94\).

$$x = 133$$

Exercise 9:

The exterior angle \(x\) is the sum of the two non - adjacent interior angles. So \(x=43 + 81\).

$$x=124$$

Exercise 10:

First, find the non - labeled interior angle of the smaller triangle. Let's call it \(y\). The angle adjacent to \(81^{\circ}\) inside the smaller triangle is \(180 - 81=99^{\circ}\). Then, using the fact that the sum of angles in a triangle is \(180^{\circ}\) in the smaller triangle, \(y=180-(99 + 19)=62^{\circ}\). Now, \(x\) is an exterior angle of the larger triangle with non - adjacent interior angles \(y\) and \(81^{\circ}\). So \(x=y + 81=62+81 = 143\)

Exercise 11:

The exterior angle \(x\) is the sum of the two non - adjacent interior angles. First, find the angle adjacent to \(115^{\circ}\) which is \(180 - 115 = 65^{\circ}\). Then \(x=65+41\).

$$x=106$$

Exercise 12:

The exterior angle \(x\) is the sum of the two non - adjacent interior angles. So \(x=41+36\).

$$x = 77$$

Answer:

Exercise 7: \(x = 108\)
Exercise 8: \(x = 133\)
Exercise 9: \(x = 124\)
Exercise 10: \(x = 143\)
Exercise 11: \(x = 106\)
Exercise 12: \(x = 77\)