Question

using angle relationships to find angle measures
directions: find the missing measures in each figure. keep the angle relationships in mind.
1.
2.
3.
4.
5.

1. ∠1 and ∠2 are vertical angles. if the measure of ∠2 is 105°, find the measure of ∠1.
2. ∠a and ∠b are complementary angles. if the measure of ∠a is 42°, find the measure of ∠b.
3. ∠p and ∠q are supplementary angles. if the measure of ∠q is 64°, find the measure of ∠p.
4. ∠1 and ∠2 form a linear pair. if the measure of ∠1 is 113°, find the measure of ∠2.

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are equal. In problem 1, $x = 112^{\circ}$ since vertical angles have the same measure.

Step2: Identify complementary - angle relationship

Complementary angles add up to $90^{\circ}$. In problem 2, $x=90 - 68=22^{\circ}$.

Step3: Identify supplementary - angle relationship

Supplementary angles add up to $180^{\circ}$. In problem 3, $x = 180 - 124=56^{\circ}$.

Step4: Use vertical - angle and supplementary - angle relationships in problem 4

$x = 43^{\circ}$ (vertical angles), $y=180 - 43 = 137^{\circ}$ (supplementary to $x$), $z = 137^{\circ}$ (vertical to $y$).

Step5: Use right - angle, vertical - angle and supplementary - angle relationships in problem 5

$x = 72^{\circ}$ (vertical angles), $y = 90^{\circ}$ (right - angle), $z=180-(90 + 72)=18^{\circ}$.

Step6: Use vertical - angle property in problem 6

Since $\angle1$ and $\angle2$ are vertical angles, $\angle1=105^{\circ}$.

Step7: Use complementary - angle property in problem 7

Since $\angle A$ and $\angle B$ are complementary, $\angle B=90 - 42 = 48^{\circ}$.

Step8: Use supplementary - angle property in problem 8

Since $\angle P$ and $\angle Q$ are supplementary, $\angle P=180 - 64=116^{\circ}$.

Step9: Use linear - pair (supplementary) property in problem 9

Since $\angle1$ and $\angle2$ form a linear pair, $\angle2=180 - 113 = 67^{\circ}$.

Answer:

1. $x = 112^{\circ}$
2. $x = 22^{\circ}$
3. $x = 56^{\circ}$
4. $x = 43^{\circ}$, $y = 137^{\circ}$, $z = 137^{\circ}$
5. $x = 72^{\circ}$, $y = 90^{\circ}$, $z = 18^{\circ}$
6. $\angle1 = 105^{\circ}$
7. $\angle B = 48^{\circ}$
8. $\angle P = 116^{\circ}$
9. $\angle2 = 67^{\circ}$