Question

in exercises 1 - 4, find m∠1 and m∠2. tell which theorem you use in each case. 1. 2. 3. 4.

Explanation:

Step1: Identify vertical - angles

Vertical angles are equal. In each of the given figures, ∠1 and the given angle are vertical angles, and ∠2 and the given angle are either vertical angles or supplementary to vertical - angle pairs.

Step2: For Figure 1

The given angle is 117°. Since ∠1 and the 117° angle are vertical angles, $m\angle1 = 117^{\circ}$. ∠2 and the 117° angle are supplementary (linear - pair). So, $m\angle2=180 - 117=63^{\circ}$. The theorems used are the Vertical Angles Theorem and the Linear - Pair Postulate.

Step3: For Figure 2

The given angle is 150°. Since ∠1 and the 150° angle are vertical angles, $m\angle1 = 150^{\circ}$. ∠2 and the 150° angle are supplementary. So, $m\angle2=180 - 150 = 30^{\circ}$. The theorems used are the Vertical Angles Theorem and the Linear - Pair Postulate.

Step4: For Figure 3

The given angle is 122°. Since ∠1 and the 122° angle are vertical angles, $m\angle1 = 122^{\circ}$. ∠2 and the 122° angle are supplementary. So, $m\angle2=180 - 122=58^{\circ}$. The theorems used are the Vertical Angles Theorem and the Linear - Pair Postulate.

Step5: For Figure 4

The given angle is 140°. Since ∠1 and the 140° angle are vertical angles, $m\angle1 = 140^{\circ}$. ∠2 and the 140° angle are supplementary. So, $m\angle2=180 - 140 = 40^{\circ}$. The theorems used are the Vertical Angles Theorem and the Linear - Pair Postulate.

Answer:

Figure 1: $m\angle1 = 117^{\circ}$, $m\angle2=63^{\circ}$ (Vertical Angles Theorem and Linear - Pair Postulate)
Figure 2: $m\angle1 = 150^{\circ}$, $m\angle2 = 30^{\circ}$ (Vertical Angles Theorem and Linear - Pair Postulate)
Figure 3: $m\angle1 = 122^{\circ}$, $m\angle2=58^{\circ}$ (Vertical Angles Theorem and Linear - Pair Postulate)
Figure 4: $m\angle1 = 140^{\circ}$, $m\angle2 = 40^{\circ}$ (Vertical Angles Theorem and Linear - Pair Postulate)