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 15. ∠2 and ∠3
2. ∠3 and ∠4 17. ∠3 and ∠1

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

1. supplement of ∠a 19. complement of ∠a
2. supplement of ∠b 21. complement of ∠b
3. 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.
4. art in the stained glass pattern, ∠1≅∠2, ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. if m∠1 = 22.3°, find m∠2, m∠3, and m∠4.

Explanation:

Step1: Recall angle - relationship definitions

Adjacent angles share a common side and a common vertex. Linear - pair 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 angle relationships for 14 - 17

For ∠1 and ∠4:

They are not adjacent as they do not share a common side.

For ∠2 and ∠3:

They are adjacent but do not form a linear pair.

For ∠3 and ∠4:

They are adjacent and form a linear pair.

For ∠3 and ∠1:

They are not adjacent.

Step3: Calculate supplements and complements for 18 - 21

For the supplement of ∠A (m∠A = 56.4°):

The supplement of an angle θ is 180°−θ. So, 180 - 56.4=123.6°.

For the complement of ∠A:

The complement of an angle θ is 90°−θ. So, 90 - 56.4 = 33.6°.

For the supplement of ∠B (m∠B=(2x - 4)°):

The supplement is 180-(2x - 4)=184 - 2x.

For the complement of ∠B:

The complement is 90-(2x - 4)=94 - 2x.

Step4: Solve for the angle and its complement in 22

Let the angle be x and its complement be y. We know that x = 3y and x + y=90. Substitute x = 3y into x + y=90, we get 3y + y=90, 4y=90, y = 22.5°, and x = 67.5°.

Step5: Find angle measures in 23

Since ∠1≅∠2 and m∠1 = 22.3°, then m∠2 = 22.3°.
Since ∠1 and ∠3 are complementary, m∠3=90 - 22.3 = 67.7°.
Since ∠2 and ∠4 are complementary, m∠4=90 - 22.3 = 67.7°.

Answer:

1. Not adjacent
2. Adjacent only
3. Adjacent and form a linear pair
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 = 22.3°, m∠3 = 67.7°, m∠4 = 67.7°