Question

in problems 1 - 4, use the diagrams. 1. name a pair of adjacent complementary angles. 2. name a pair of adjacent supplementary angles. 3. name a pair of nonadjacent supplementary angles. 4. name a pair of nonadjacent complementary angles. 5. ∠1 is a complement of ∠2, and m∠1 = 23°. find m∠2. 6. ∠3 is a complement of ∠4, and m∠3 = 46°. find m∠4. 7. ∠5 is a supplement of ∠6, and m∠5 = 78°. find m∠6. 8. ∠7 is a supplement of ∠8, and m∠7 = 109°. find m∠8. 9. find the measure of each angle (3x + 5)° (10x - 7)°. 10. find the measure of each angle (15x - 2)° (7x + 4)°. 11. ∠uvw and ∠xyz are complementary angles. m∠uvw=(x - 10)°, and m∠xyz=(4x - 10)°. 12. ∠efg and ∠lmn are supplementary angles. m∠efg=(3x + 17)°, and m∠lmn=($\frac{1}{2}$x - 5)°

Explanation:

Step1: Recall angle - relationship definitions

Complementary angles add up to 90°, and supplementary angles add up to 180°. Adjacent angles share a common side and a common vertex.

Step2: Solve problem 1

In the diagram, adjacent complementary angles are angles that share a side and sum to 90°. For example, if we consider the angles around a right - angled intersection, we can find such pairs. But without a specific diagram reference, assume in a right - angled situation, if one angle is 34° and another is 56°, they are adjacent complementary angles.

Step3: Solve problem 2

Adjacent supplementary angles share a side and sum to 180°. For example, two angles on a straight line are adjacent supplementary angles.

Step4: Solve problem 3

Non - adjacent supplementary angles do not share a side but still sum to 180°. In a more complex geometric figure with multiple lines and angles, we can find such pairs.

Step5: Solve problem 4

Non - adjacent complementary angles do not share a side but sum to 90°.

Step6: Solve problem 5

If ∠1 is a complement of ∠2 and m∠1 = 23°, then m∠2=90° - m∠1. So, m∠2 = 90°−23° = 67°.

Step7: Solve problem 6

If ∠3 is a complement of ∠4 and m∠3 = 46°, then m∠4=90° - m∠3. So, m∠4 = 90°−46° = 44°.

Step8: Solve problem 7

If ∠5 is a supplement of ∠6 and m∠5 = 78°, then m∠6=180° - m∠5. So, m∠6 = 180°−78° = 102°.

Step9: Solve problem 8

If ∠7 is a supplement of ∠8 and m∠7 = 109°, then m∠8=180° - m∠7. So, m∠8 = 180°−109° = 71°.

Step10: Solve problem 9

If (3x + 5)° and (10x−7)° are supplementary (since they are on a straight line), then (3x + 5)+(10x−7)=180. Combine like terms: 13x−2 = 180. Add 2 to both sides: 13x=182. Divide by 13: x = 14. Then the angles are 3x + 5=3(14)+5 = 47° and 10x−7=10(14)−7 = 133°.

Step11: Solve problem 10

If (15x−2)° and (7x + 4)° are complementary, then (15x−2)+(7x + 4)=90. Combine like terms: 22x+2 = 90. Subtract 2 from both sides: 22x=88. Divide by 22: x = 4. Then the angles are 15x−2=15(4)−2 = 58° and 7x + 4=7(4)+4 = 32°.

Step12: Solve problem 11

If ∠UVW and ∠XYZ are complementary, then (x - 10)+(4x−10)=90. Combine like terms: 5x−20 = 90. Add 20 to both sides: 5x=110. Divide by 5: x = 22. Then m∠UVW=x - 10=22 - 10 = 12° and m∠XYZ=4x−10=4(22)−10 = 78°.

Step13: Solve problem 12

If ∠EFG and ∠LMN are supplementary, then (3x + 17)+(1/2x−5)=180. First, get a common denominator for the x - terms: (6x/2+17)+(x/2−5)=180. Combine like terms: (6x + x)/2+(17 - 5)=180, or 7x/2+12 = 180. Subtract 12 from both sides: 7x/2=168. Multiply both sides by 2/7: x = 48. Then m∠EFG=3x + 17=3(48)+17 = 161° and m∠LMN=1/2x−5=1/2(48)−5 = 19°.

Answer:

1. Example: 34° and 56° adjacent complementary angles (depending on diagram)
2. Example: Angles on a straight - line adjacent supplementary angles
3. Non - adjacent angles summing to 180°
4. Non - adjacent angles summing to 90°
5. m∠2 = 67°
6. m∠4 = 44°
7. m∠6 = 102°
8. m∠8 = 71°
9. 47° and 133°
10. 58° and 32°
11. m∠UVW = 12°, m∠XYZ = 78°
12. m∠EFG = 161°, m∠LMN = 19°