Question

1. find the missing measure.
2. find the missing measure.
3. find the missing measures
4. if the measure of an angle is 13°, find the measure of its supplement.
5. if the measure of an angle is 38°, find the measure of its complement.
6. ∠1 and ∠2 form a linear - pair. if m∠1=(5x + 9)° and m∠2=(3x + 11)°, find the measure of each angle.
7. ∠1 and ∠2 are vertical angles. if m∠1=(17x + 1)° and m∠2=(20x - 14)°, find m∠2
8. ∠k and ∠l are complementary angles. if m∠k=(3x + 3)° and m∠l=(10x - 4)°, find the measure of each angle.
9. if m∠p is three less than twice the measure of ∠q, and ∠p and ∠q are supplementary angles, find each angle measure.
10. if m∠b is two more than three times the measure of ∠c, and ∠b and ∠c are complementary angles, find each angle measure.

Explanation:

Step1: Recall angle - addition property

For question 1, if the two angles are adjacent and form a right - angle (assuming the unlabeled angle is part of a right - angle), and one angle is 65°, then \(x + 65=90\). Solving for \(x\), we get \(x=90 - 65\).

Step2: Calculate \(x\)

\(x = 25\)

Step3: Recall vertical - angle property

For question 2, vertical angles are equal. If one vertical angle is 51°, then \(x = 51\)

Step4: Recall vertical and supplementary angle properties

For question 3, vertical angles are equal, so \(x = 107\). Also, since \(x\) and \(y\) are supplementary (\(x + y=180\)), then \(y=180 - 107=73\), and \(z=x = 107\)

Step5: Recall supplementary angle definition

For question 4, the supplement of an angle \(\theta\) is \(180-\theta\). If \(\theta = 13\), then the supplement is \(180 - 13=167\)

Step6: Recall complementary angle definition

For question 5, the complement of an angle \(\theta\) is \(90-\theta\). If \(\theta = 38\), then the complement is \(90 - 38 = 52\)

Step7: Recall linear - pair property

For question 6, since \(\angle1\) and \(\angle2\) form a linear pair, \(m\angle1+m\angle2 = 180\). So \((5x + 9)+(3x + 11)=180\). Combine like terms: \(8x+20 = 180\), then \(8x=160\), and \(x = 20\). So \(m\angle1=5x + 9=5\times20 + 9=109\) and \(m\angle2=3x + 11=3\times20+11 = 71\)

Step8: Recall vertical - angle property

For question 7, since \(\angle1\) and \(\angle2\) are vertical angles, \(m\angle1=m\angle2\). So \(17x + 1=20x-14\). Subtract \(17x\) from both sides: \(1 = 3x-14\). Add 14 to both sides: \(15 = 3x\), then \(x = 5\). So \(m\angle2=20x-14=20\times5-14 = 86\)

Step9: Recall complementary - angle property

For question 8, since \(\angle K\) and \(\angle L\) are complementary, \(m\angle K+m\angle L=90\). So \((3x + 3)+(10x-4)=90\). Combine like terms: \(13x-1 = 90\), then \(13x=91\), and \(x = 7\). So \(m\angle K=3x + 3=3\times7+3 = 24\) and \(m\angle L=10x-4=10\times7-4 = 66\)

Step10: Set up equations for supplementary angles

For question 9, let \(m\angle Q=x\), then \(m\angle P=2x - 3\). Since \(\angle P\) and \(\angle Q\) are supplementary, \(m\angle P+m\angle Q=180\). So \((2x - 3)+x=180\). Combine like terms: \(3x-3 = 180\), then \(3x=183\), and \(x = 61\). So \(m\angle Q=61\) and \(m\angle P=2\times61-3=119\)

Step11: Set up equations for complementary angles

For question 10, let \(m\angle C=x\), then \(m\angle B=3x + 2\). Since \(\angle B\) and \(\angle C\) are complementary, \(m\angle B+m\angle C=90\). So \((3x + 2)+x=90\). Combine like terms: \(4x+2 = 90\), then \(4x=88\), and \(x = 22\). So \(m\angle C=22\) and \(m\angle B=3\times22 + 2=68\)

Answer:

1. \(x = 25\)
2. \(x = 51\)
3. \(x = 107\), \(y = 73\), \(z = 107\)
4. \(167\)
5. \(52\)
6. \(m\angle1 = 109\), \(m\angle2 = 71\)
7. \(m\angle2 = 86\)
8. \(m\angle K=24\), \(m\angle L=66\)
9. \(m\angle P=119\), \(m\angle Q=61\)
10. \(m\angle B=68\), \(m\angle C=22\)