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.

Explanation:

Step1: Recall angle - related properties

For question 1, assume the angles are adjacent and form a right - angle (90°). Then $x + 65=90$.

Step2: Solve for $x$ in question 1

$x=90 - 65=25$.

Step3: Recall vertical - angle property for question 2

Vertical angles are equal. So if one angle is 51°, then $x = 51$.

Step4: Recall supplementary and vertical - angle properties for question 3

Vertical angles: $x = 107$. Supplementary angles: $y+107 = 180$, so $y=180 - 107 = 73$. And $z=x = 107$ (vertical angles).

Step5: Recall supplementary - angle property 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 property 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: $8x+20 = 180$. Subtract 20 from both sides: $8x=160$. Divide by 8: $x = 20$. Then $m\angle1=5x + 9=5\times20+9 = 109$ 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: $15 = 3x$. Divide by 3: $x = 5$. Then $m\angle2=20x-14=20\times5-14 = 86$.

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: $13x-1 = 90$. Add 1 to both sides: $13x=91$. Divide by 13: $x = 7$. Then $m\angle K=3x + 3=3\times7+3 = 24$ and $m\angle L=10x-4=10\times7-4 = 66$.

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. $m\angle K = 24$, $m\angle L = 66$