Question

1. 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: Set up equation for problem 9

Let $m\angle Q=x$ and $m\angle P = 2x - 3$. Since $\angle P$ and $\angle Q$ are supplementary, $m\angle P+m\angle Q=180^{\circ}$. So, $(2x - 3)+x=180$.

Step2: Combine like - terms

Combining the $x$ terms on the left - hand side gives $3x-3 = 180$.

Step3: Add 3 to both sides

Adding 3 to both sides of the equation: $3x-3 + 3=180 + 3$, which simplifies to $3x=183$.

Step4: Solve for x

Dividing both sides by 3: $x=\frac{183}{3}=61$.

Step5: Find angle measures

$m\angle Q=x = 61^{\circ}$, $m\angle P=2x - 3=2\times61-3=122 - 3=119^{\circ}$.

Step6: Set up equation for problem 10

Let $m\angle C=x$. Then $m\angle B = 3x+2$. Since $\angle B$ and $\angle C$ are complementary, $m\angle B+m\angle C = 90^{\circ}$. So, $(3x + 2)+x=90$.

Step7: Combine like - terms

Combining the $x$ terms on the left - hand side gives $4x+2 = 90$.

Step8: Subtract 2 from both sides

$4x+2-2=90 - 2$, which simplifies to $4x=88$.

Step9: Solve for x

Dividing both sides by 4: $x=\frac{88}{4}=22$.

Step10: Find angle measures

$m\angle C=x = 22^{\circ}$, $m\angle B=3x + 2=3\times22+2=66 + 2=68^{\circ}$.

Answer:

For problem 9: $m\angle P = 119^{\circ}$, $m\angle Q=61^{\circ}$
For problem 10: $m\angle B = 68^{\circ}$, $m\angle C=22^{\circ}$