Question

practice and problem solving
tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent.

1. ∠1 and ∠4
2. ∠2 and ∠3
3. ∠3 and ∠4
4. ∠2 and ∠1

given m∠a = 56.4° and m∠b=(2x - 4)°, find the measure of each of the following.

1. supplement of ∠a
2. complement of ∠a
3. supplement of ∠b
4. complement of ∠b
5. multi - step an angles measure is 3 times the measure of its complement. find the measure of the angle and the measure of its complement.
6. art in the stained glass pattern, ∠1≅∠2. ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. if m∠1 = 22.5°, find m∠2, m∠3, and m∠4.

Explanation:

Step1: Recall angle - relationship definitions

Adjacent angles share a common side and a common vertex. A linear - pair of angles are adjacent and their non - common sides form a straight line. Complementary angles add up to 90° and supplementary angles add up to 180°.

Step2: Analyze problem 14

For ∠1 and ∠4, they are not adjacent as they do not share a common side.

Step3: Analyze problem 15

For ∠2 and ∠3, they are adjacent as they share a common side and a common vertex, but they do not form a linear pair as their non - common sides do not form a straight line.

Step4: Analyze problem 16

For ∠3 and ∠4, they are not adjacent as they do not share a common side.

Step5: Analyze problem 17

For ∠2 and ∠1, they are not adjacent as they do not share a common side.

Step6: Given m∠A = 56.4°

The supplement of ∠A is 180°−56.4° = 123.6°.

Step7: Given m∠B=(2x - 4)°

The supplement of ∠B is 180°-(2x - 4)°=(184 - 2x)°.

Step8: Complement of ∠A

The complement of ∠A is 90°−56.4° = 33.6°.

Step9: Complement of ∠B

The complement of ∠B is 90°-(2x - 4)°=(94 - 2x)°.

Step10: For problem 22

Let the measure of the angle be x. Its complement is 90 - x. Given x = 3(90 - x). Expand: x=270 - 3x. Add 3x to both sides: 4x = 270. So x = 67.5°. Its complement is 90 - 67.5 = 22.5°.

Step11: For problem 23

Since ∠1≅∠2 and m∠1 = 22.5°, then m∠2 = 22.5°. Since ∠1 and ∠3 are complementary, m∠3=90° - 22.5° = 67.5°. Since ∠2 and ∠4 are complementary, m∠4=90° - 22.5° = 67.5°.

Answer:

1. Not adjacent
2. Adjacent, not a linear pair
3. Not adjacent
4. Not adjacent
5. 123.6°
6. 33.6°
7. (184 - 2x)°
8. (94 - 2x)°
9. Angle: 67.5°, Complement: 22.5°
10. m∠2 = 22.5°, m∠3 = 67.5°, m∠4 = 67.5°