Question

1. given m∠xyz=(4x - 15)° and m∠stu=(2x + 3)°. find the value of x and the measure of each angle.

part a
∠xyz and ∠stu are supplementary.
part b
∠xyz and ∠stu are congruent.
part c
∠xyz and ∠stu are complementary.

Explanation:

Step1: Recall supplementary - angle property

Supplementary angles sum to 180°. So, $(4x - 15)+(2x + 3)=180$.

Step2: Combine like - terms

$4x+2x-15 + 3=180$, which simplifies to $6x-12 = 180$.

Step3: Isolate the variable term

Add 12 to both sides: $6x=180 + 12$, so $6x=192$.

Step4: Solve for x

Divide both sides by 6: $x=\frac{192}{6}=32$.

Step5: Find the measure of each angle

For $\angle XYZ=4x - 15$, substitute $x = 32$: $4\times32-15=128 - 15 = 113^{\circ}$.
For $\angle STU=2x + 3$, substitute $x = 32$: $2\times32+3=64 + 3 = 67^{\circ}$.

For Part B:

Step1: Recall congruent - angle property

Congruent angles are equal. So, $4x-15=2x + 3$.

Step2: Isolate the variable terms

Subtract $2x$ from both sides: $4x-2x-15=2x-2x + 3$, which gives $2x-15 = 3$.

Step3: Isolate the variable

Add 15 to both sides: $2x=3 + 15$, so $2x=18$.

Step4: Solve for x

Divide both sides by 2: $x = 9$.

Step5: Find the measure of each angle

For $\angle XYZ=4x - 15$, substitute $x = 9$: $4\times9-15=36 - 15 = 21^{\circ}$.
For $\angle STU=2x + 3$, substitute $x = 9$: $2\times9+3=18 + 3 = 21^{\circ}$.

For Part C:

Step1: Recall complementary - angle property

Complementary angles sum to 90°. So, $(4x-15)+(2x + 3)=90$.

Step2: Combine like - terms

$4x+2x-15 + 3=90$, which simplifies to $6x-12 = 90$.

Step3: Isolate the variable term

Add 12 to both sides: $6x=90 + 12$, so $6x=102$.

Step4: Solve for x

Divide both sides by 6: $x=\frac{102}{6}=17$.

Step5: Find the measure of each angle

For $\angle XYZ=4x - 15$, substitute $x = 17$: $4\times17-15=68 - 15 = 53^{\circ}$.
For $\angle STU=2x + 3$, substitute $x = 17$: $2\times17+3=34 + 3 = 37^{\circ}$.

Answer:

Part A:
$x = 32$
$\angle XYZ=113^{\circ}$
$\angle STU=67^{\circ}$

Part B:
$x = 9$
$\angle XYZ=21^{\circ}$
$\angle STU=21^{\circ}$

Part C:
$x = 17$
$\angle XYZ=53^{\circ}$
$\angle STU=37^{\circ}$