Question

classify △abc by angles and sides. angles (circle one): acute, right, obtuse. sides (circle one): isosceles, scalene, equilateral. classify △dec by angles and sides. angles (circle one): acute, right, obtuse. sides (circle one): isosceles, scalene, equilateral. given m∠xyz=(8x + 6)° and m∠stu=(7x + 9)°, find the value of x and the measure of each angle. part a: ∠xyz and ∠stu are congruent. part b: ∠xyz and ∠stu are complementary. part c: ∠xyz and ∠stu are supplementary.

Explanation:

Step1: Set up equation for congruent angles

Since $\angle XYZ$ and $\angle STU$ are congruent, we set $8x + 6=7x + 9$.
$8x+6 = 7x + 9$

Step2: Solve for $x$

Subtract $7x$ from both sides: $8x-7x+6=7x - 7x+9$, so $x=3$.
$x = 3$

Step3: Find measure of angles

For $\angle XYZ$, substitute $x = 3$ into $8x + 6$: $8\times3+6=24 + 6=30^{\circ}$.
$\angle XYZ=30^{\circ}$
For $\angle STU$, substitute $x = 3$ into $7x + 9$: $7\times3+9=21 + 9=30^{\circ}$.
$\angle STU=30^{\circ}$

For part B:

Step1: Set up equation for complementary angles

Since $\angle XYZ$ and $\angle STU$ are complementary, $(8x + 6)+(7x + 9)=90$.
$(8x + 6)+(7x + 9)=90$

Step2: Combine like - terms

$8x+7x+6 + 9=90$, so $15x+15 = 90$.
$15x+15=90$

Step3: Isolate the variable term

Subtract 15 from both sides: $15x+15-15=90 - 15$, so $15x=75$.
$15x=75$

Step4: Solve for $x$

Divide both sides by 15: $x = 5$.
$x = 5$

Step5: Find measure of angles

For $\angle XYZ$, substitute $x = 5$ into $8x + 6$: $8\times5+6=40+6 = 46^{\circ}$.
$\angle XYZ=46^{\circ}$
For $\angle STU$, substitute $x = 5$ into $7x + 9$: $7\times5+9=35 + 9=44^{\circ}$.
$\angle STU=44^{\circ}$

For part C:

Step1: Set up equation for supplementary angles

Since $\angle XYZ$ and $\angle STU$ are supplementary, $(8x + 6)+(7x + 9)=180$.
$(8x + 6)+(7x + 9)=180$

Step2: Combine like - terms

$8x+7x+6 + 9=180$, so $15x+15 = 180$.
$15x+15=180$

Step3: Isolate the variable term

Subtract 15 from both sides: $15x+15-15=180 - 15$, so $15x=165$.
$15x=165$

Step4: Solve for $x$

Divide both sides by 15: $x = 11$.
$x = 11$

Step5: Find measure of angles

For $\angle XYZ$, substitute $x = 11$ into $8x + 6$: $8\times11+6=88+6 = 94^{\circ}$.
$\angle XYZ=94^{\circ}$
For $\angle STU$, substitute $x = 11$ into $7x + 9$: $7\times11+9=77 + 9=86^{\circ}$.
$\angle STU=86^{\circ}$

Answer:

Part A:
$x = 3$
$\angle XYZ=30^{\circ}$
$\angle STU=30^{\circ}$
Part B:
$x = 5$
$\angle XYZ=46^{\circ}$
$\angle STU=44^{\circ}$
Part C:
$x = 11$
$\angle XYZ=94^{\circ}$
$\angle STU=86^{\circ}$