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
10. find the measure of each angle
11. ∠uvw and ∠xyz are complementary angles. m∠uvw=(x - 10)°, and m∠xyz=(4x - 10)°
12. ∠efg and ∠lmn are supplementary angle m∠efg=(3x + 17)°, and m∠lmn=($\frac{1}{2}$x - 5)°

Explanation:

Step1: Recall angle - relationship formulas

Complementary angles: \(m\angle A + m\angle B=90^{\circ}\). Supplementary angles: \(m\angle C + m\angle D = 180^{\circ}\).

Step2: Solve problem 5

Given \(\angle1\) is a complement of \(\angle2\) and \(m\angle1 = 23^{\circ}\). Using the formula \(m\angle1+m\angle2 = 90^{\circ}\), we substitute \(m\angle1\): \(23^{\circ}+m\angle2=90^{\circ}\). Then \(m\angle2=90^{\circ}- 23^{\circ}=67^{\circ}\).

Step3: Solve problem 6

Given \(\angle3\) is a complement of \(\angle4\) and \(m\angle3 = 46^{\circ}\). Using the formula \(m\angle3+m\angle4 = 90^{\circ}\), we substitute \(m\angle3\): \(46^{\circ}+m\angle4=90^{\circ}\). Then \(m\angle4=90^{\circ}-46^{\circ}=44^{\circ}\).

Step4: Solve problem 7

Given \(\angle5\) is a supplement of \(\angle6\) and \(m\angle5 = 78^{\circ}\). Using the formula \(m\angle5+m\angle6 = 180^{\circ}\), we substitute \(m\angle5\): \(78^{\circ}+m\angle6=180^{\circ}\). Then \(m\angle6=180^{\circ}-78^{\circ}=102^{\circ}\).

Step5: Solve problem 8

Given \(\angle7\) is a supplement of \(\angle8\) and \(m\angle7 = 109^{\circ}\). Using the formula \(m\angle7+m\angle8 = 180^{\circ}\), we substitute \(m\angle7\): \(109^{\circ}+m\angle8=180^{\circ}\). Then \(m\angle8=180^{\circ}-109^{\circ}=71^{\circ}\).

Step6: Solve problem 9

The angles \((3x + 5)^{\circ}\) and \((10x-7)^{\circ}\) are supplementary. So \((3x + 5)+(10x-7)=180\). Combine like - terms: \(13x-2 = 180\). Add 2 to both sides: \(13x=182\). Divide both sides by 13: \(x = 14\). Then \(3x + 5=3\times14 + 5=42 + 5=47^{\circ}\) and \(10x-7=10\times14-7=140 - 7=133^{\circ}\).

Step7: Solve problem 10

The angles \((15x-2)^{\circ}\) and \((7x + 4)^{\circ}\) are complementary. So \((15x-2)+(7x + 4)=90\). Combine like - terms: \(22x+2 = 90\). Subtract 2 from both sides: \(22x=88\). Divide both sides by 22: \(x = 4\). Then \(15x-2=15\times4-2=60 - 2=58^{\circ}\) and \(7x + 4=7\times4+4=28 + 4=32^{\circ}\).

Step8: Solve problem 11

Since \(\angle UVW\) and \(\angle XYZ\) are complementary, \((x - 10)+(4x-10)=90\). Combine like - terms: \(5x-20 = 90\). Add 20 to both sides: \(5x=110\). Divide both sides by 5: \(x = 22\). Then \(m\angle UVW=x - 10=22-10 = 12^{\circ}\) and \(m\angle XYZ=4x-10=4\times22-10=88 - 10=78^{\circ}\).

Step9: Solve problem 12

Since \(\angle EFG\) and \(\angle LMN\) are supplementary, \((3x + 17)+(\frac{1}{2}x-5)=180\). Combine like - terms: \(3x+\frac{1}{2}x+17 - 5=180\), \(\frac{6x + x}{2}+12=180\), \(\frac{7x}{2}=168\). Multiply both sides by \(\frac{2}{7}\): \(x = 48\). Then \(m\angle EFG=3x + 17=3\times48+17=144 + 17=161^{\circ}\) and \(m\angle LMN=\frac{1}{2}x-5=\frac{1}{2}\times48-5=24 - 5=19^{\circ}\).

Answer:

1. \(m\angle2 = 67^{\circ}\)
2. \(m\angle4 = 44^{\circ}\)
3. \(m\angle6 = 102^{\circ}\)
4. \(m\angle8 = 71^{\circ}\)
5. The angles are \(47^{\circ}\) and \(133^{\circ}\)
6. The angles are \(58^{\circ}\) and \(32^{\circ}\)
7. \(m\angle UVW = 12^{\circ}\), \(m\angle XYZ = 78^{\circ}\)
8. \(m\angle EFG = 161^{\circ}\), \(m\angle LMN = 19^{\circ}\)