Question

directions: find all missing angles.
1.
2.
3.
4.
5.
6.
7.
directions: find the value of x.
8.
9.
10.
11.

Explanation:

Step1: Use the triangle angle - sum theorem (the sum of angles in a triangle is \(180^{\circ}\)) for problem 1

For the first triangle, if two angles are \(76^{\circ}\) and \(59^{\circ}\), then \(m\angle1=180-(76 + 59)\)

$$m\angle1=180 - 135=45^{\circ}$$

Step2: Use the exterior - angle theorem (an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles) for problem 2

The exterior angle \(m\angle1\) is equal to \(62+67\)

$$m\angle1 = 129^{\circ}$$

Step3: Use the exterior - angle theorem for problem 3

First, find the adjacent interior angle to \(152^{\circ}\), which is \(180 - 152=28^{\circ}\). Then, using the triangle angle - sum theorem in the triangle with angles \(28^{\circ}\), \(115^{\circ}\), and \(\angle1\), \(m\angle1=180-(28 + 115)\)

$$m\angle1=180-143 = 37^{\circ}$$

Step4: Use the vertical - angles and triangle angle - sum theorems for problem 4

• For \(\angle2\): Since \(\angle2\) and \(42^{\circ}\) are vertical angles, \(m\angle2 = 42^{\circ}\)
• For \(\angle1\): Using the triangle angle - sum theorem in the left - hand triangle, \(m\angle1=180-(50 + 42)\)

$$m\angle1=180 - 92=88^{\circ}$$

• For \(\angle3\): Using the triangle angle - sum theorem in the right - hand triangle, \(m\angle3=180-(25 + 42)\)

$$m\angle3=180 - 67 = 113^{\circ}$$

Step5: Use the exterior - angle and triangle angle - sum theorems for problem 5

• For \(\angle1\): \(m\angle1=180 - 118=62^{\circ}\)
• For \(\angle2\): Using the triangle angle - sum theorem in the large left - hand triangle, if one angle is \(62^{\circ}\) and another is \(73^{\circ}\), then the non - labeled angle is \(180-(62 + 73)=45^{\circ}\). Then, using the exterior - angle theorem for the small triangle with \(49^{\circ}\) exterior angle, \(m\angle2=49-(45)=4^{\circ}\)
• For \(\angle3\): \(m\angle3=180-(4)=176^{\circ}\)

Answer:

1. \(m\angle1 = 45^{\circ}\)
2. \(m\angle1=129^{\circ}\)
3. \(m\angle1 = 37^{\circ}\)
4. \(m\angle1 = 88^{\circ}\), \(m\angle2 = 42^{\circ}\), \(m\angle3=113^{\circ}\)
5. \(m\angle1 = 62^{\circ}\), \(m\angle2 = 4^{\circ}\), \(m\angle3=176^{\circ}\)