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.

1. using the diagram below, find all angles congruent to the given angle given ab||cd.

Explanation:

Part A

Step1: Set up equation for supplementary angles

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

Step2: Combine like - terms

Combine the $x$ terms and the constant terms: $(4x + 2x)+(-15 + 3)=180$, which simplifies to $6x-12 = 180$.

Step3: Isolate the variable term

Add 12 to both sides of the equation: $6x-12 + 12=180 + 12$, resulting in $6x=192$.

Step4: Solve for x

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

Step5: Find the measure of ∠XYZ

Substitute $x = 32$ into the expression for ∠XYZ: $m\angle XYZ=4x-15=4\times32-15=128 - 15=113^{\circ}$.

Step6: Find the measure of ∠STU

Substitute $x = 32$ into the expression for ∠STU: $m\angle STU=2x + 3=2\times32+3=64 + 3=67^{\circ}$.

Part B

Step1: Set up equation for congruent angles

Since ∠XYZ and ∠STU are congruent, $4x-15=2x + 3$.

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: $2x-15 + 15=3 + 15$, resulting in $2x=18$.

Step4: Solve for x

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

Step5: Find the measure of ∠XYZ

Substitute $x = 9$ into the expression for ∠XYZ: $m\angle XYZ=4x-15=4\times9-15=36-15=21^{\circ}$.

Step6: Find the measure of ∠STU

Since they are congruent, $m\angle STU=21^{\circ}$.

Part C

Step1: Set up equation for complementary angles

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: $6x-12 + 12=90 + 12$, resulting in $6x=102$.

Step4: Solve for x

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

Step5: Find the measure of ∠XYZ

Substitute $x = 17$ into the expression for ∠XYZ: $m\angle XYZ=4x-15=4\times17-15=68-15=53^{\circ}$.

Step6: Find the measure of ∠STU

Substitute $x = 17$ into the expression for ∠STU: $m\angle STU=2x + 3=2\times17+3=34 + 3=37^{\circ}$.

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

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