Question

determine whether the labeled angles are vertical angles, a linear pair, or neither.
use the linear pair postulate to find the measure of ∠2.
use the vertical angles theorem to find an angle that is congruent to ∠1.
use the vertical angles theorem and the linear pair postulate to find m∠1, m∠2, and m∠3.
find the value of x.
the window - frame shown at the right forms angles 1, 2, 3, and 4. the measure of ∠1 is 70°.

1. name two pairs of vertical angles.
2. find m∠2.
3. find m∠3.
4. find m∠4.

Explanation:

Step1: Recall angle - pair definitions

Vertical angles are opposite angles formed by two intersecting lines. A linear pair is a pair of adjacent angles whose non - common sides are opposite rays and their sum is 180°.

Step2: Analyze problem 3

Angles 1 and 2 are vertical angles since they are opposite angles formed by two intersecting lines.

Step3: Analyze problem 4

Angles 3 and 4 are neither vertical angles nor a linear pair as they are not formed by two intersecting lines in the required way.

Step4: Analyze problem 5

Angles 5 and 6 are vertical angles as they are opposite angles formed by two intersecting lines.

Step5: Analyze problem 6

Using the linear - pair postulate, if one angle is 45°, then ∠2 = 180°−45° = 135° since they form a linear pair.

Step6: Analyze problem 7

If one angle is 85°, then ∠2=180° - 85°=95° as they form a linear pair.

Step7: Analyze problem 8

If one angle is 30°, then ∠2 = 180°−30° = 150° as they form a linear pair.

Step8: Analyze problem 9

The angle congruent to ∠1 is ∠3 by the vertical - angles theorem.

Step9: Analyze problem 10

The angle congruent to ∠1 is ∠3 by the vertical - angles theorem.

Step10: Analyze problem 11

The angle congruent to ∠1 is ∠3 by the vertical - angles theorem.

Step11: Analyze problem 12

Since ∠1 and the 150° angle are vertical angles, m∠1 = 150°. ∠1 and ∠2 form a linear pair, so m∠2=180° - 150° = 30°. ∠2 and ∠3 are vertical angles, so m∠3 = 30°.

Step12: Analyze problem 13

Since ∠1 and the 54° angle are vertical angles, m∠1 = 54°. ∠1 and ∠2 form a linear pair, so m∠2=180° - 54° = 126°. ∠2 and ∠3 are vertical angles, so m∠3 = 126°.

Step13: Analyze problem 14

Since ∠1 and the 45° angle are vertical angles, m∠1 = 45°. ∠1 and ∠2 form a linear pair, so m∠2=180° - 45° = 135°. ∠2 and ∠3 are vertical angles, so m∠3 = 135°.

Step14: Analyze problem 15

Since the 32° angle and (2x)° are vertical angles, 2x = 32, so x = 16.

Step15: Analyze problem 16

Since 68° and (x + 5)° are vertical angles, x+5 = 68, so x = 63.

Step16: Analyze problem 17

Since (5x)° and 130° are vertical angles, 5x=130, so x = 26.

Step17: Analyze problem 18

Two pairs of vertical angles are ∠1 and ∠3, ∠2 and ∠4.

Step18: Analyze problem 19

∠1 and ∠2 form a linear pair. Given m∠1 = 70°, then m∠2=180° - 70° = 110°.

Step19: Analyze problem 20

∠1 and ∠3 are vertical angles, so m∠3 = 70°.

Step20: Analyze problem 21

∠2 and ∠4 are vertical angles, so m∠4 = 110°.

Answer:

1. Vertical angles
2. Neither
3. Vertical angles
4. 135°
5. 95°
6. 150°
7. ∠3
8. ∠3
9. ∠3
10. m∠1 = 150°, m∠2 = 30°, m∠3 = 30°
11. m∠1 = 54°, m∠2 = 126°, m∠3 = 126°
12. m∠1 = 45°, m∠2 = 135°, m∠3 = 135°
13. x = 16
14. x = 63
15. x = 26
16. ∠1 and ∠3, ∠2 and ∠4
17. 110°
18. 70°
19. 110°