Question

1. ∠1 and ∠2 are vertical angles. if the measure of ∠2 is 105°, find the measure of ∠1. 7. ∠a and ∠b are complementary angles. if the measure of ∠a is 42°, find the measure of ∠b. 8. ∠p and ∠q are supplementary angles. if the measure of ∠q is 64°, find the measure of ∠p. 9. ∠1 and ∠2 form a linear pair. if the measure of ∠1 is 113°, find the measure of ∠2.

Explanation:

Step1: Recall vertical - angle property

Vertical angles are equal. So if $\angle1$ and $\angle2$ are vertical angles and $m\angle2 = 105^{\circ}$, then $m\angle1=m\angle2$.
$m\angle1 = 105^{\circ}$

Step2: Recall complementary - angle property

Complementary angles add up to $90^{\circ}$. If $\angle A$ and $\angle B$ are complementary and $m\angle A = 42^{\circ}$, then $m\angle B=90^{\circ}-m\angle A$.
$m\angle B = 90^{\circ}-42^{\circ}=48^{\circ}$

Step3: Recall supplementary - angle property

Supplementary angles add up to $180^{\circ}$. If $\angle P$ and $\angle Q$ are supplementary and $m\angle Q = 64^{\circ}$, then $m\angle P=180^{\circ}-m\angle Q$.
$m\angle P = 180^{\circ}-64^{\circ}=116^{\circ}$

Step4: Recall linear - pair property

Angles in a linear pair are supplementary. If $\angle1$ and $\angle2$ form a linear pair and $m\angle1 = 113^{\circ}$, then $m\angle2=180^{\circ}-m\angle1$.
$m\angle2 = 180^{\circ}-113^{\circ}=67^{\circ}$

Answer:

1. $105^{\circ}$
2. $48^{\circ}$
3. $116^{\circ}$
4. $67^{\circ}$