Question

1. if the measure of an angle is 13°, find the measure of its supplement.
2. if the measure of an angle is 38°, find the measure of its complement.
3. ∠1 and ∠2 form a linear pair. if m∠1=(5x + 9)° and m∠2=(3x + 11)°, find the measure of each angle.
4. ∠1 and ∠2 are vertical angles. if m∠1=(17x + 1)° and m∠2=(20x - 14)°, find m∠2.
5. ∠k and ∠l are complementary angles. if m∠k=(3x + 3)° and m∠l=(10x - 4)°, find the measure of each angle.

Explanation:

4.

Step1: Recall the definition of supplementary angles

Supplementary angles add up to 180°. Let the given angle be $\alpha = 13^{\circ}$, and its supplement be $\beta$. Then $\alpha+\beta = 180^{\circ}$.

Step2: Solve for the supplement

$\beta=180^{\circ}-\alpha$. Substitute $\alpha = 13^{\circ}$ into the formula: $\beta = 180^{\circ}- 13^{\circ}=167^{\circ}$.

Step1: Recall the definition of complementary angles

Complementary angles add up to 90°. Let the given angle be $\theta = 38^{\circ}$, and its complement be $\varphi$. Then $\theta+\varphi=90^{\circ}$.

Step2: Solve for the complement

$\varphi = 90^{\circ}-\theta$. Substitute $\theta = 38^{\circ}$ into the formula: $\varphi=90^{\circ}-38^{\circ}=52^{\circ}$.

Step1: Recall the property of linear - pair angles

Linear - pair angles add up to 180°. So, $m\angle1 + m\angle2=180^{\circ}$. Given $m\angle1=(5x + 9)^{\circ}$ and $m\angle2=(3x + 11)^{\circ}$, we have the equation $(5x + 9)+(3x + 11)=180$.

Step2: Simplify the left - hand side of the equation

$5x+9 + 3x+11=8x + 20$. So, $8x+20 = 180$.

Step3: Solve for x

Subtract 20 from both sides: $8x=180 - 20=160$. Then divide both sides by 8: $x = 20$.

Step4: Find the measure of each angle

$m\angle1=(5x + 9)^{\circ}=(5\times20+9)^{\circ}=109^{\circ}$; $m\angle2=(3x + 11)^{\circ}=(3\times20 + 11)^{\circ}=71^{\circ}$.

Answer:

$167^{\circ}$

5.