Question

in exercises 9 - 12, find the measure of each angle. (see example 3.) 9. 10. 11. ∠uvw and ∠xyz are complementary angles, m∠uvw=(x - 10)°, and m∠xyz=(4x - 10)°. 12. ∠efg and ∠lmn are supplementary angles, m∠efg=(3x + 17)°, and m∠lmn=(1/2x - 5)°.

Explanation:

9.

Step1: Set up the equation

Since the two angles $(3x + 5)^{\circ}$ and $(10x-7)^{\circ}$ are supplementary (they form a straight - line and the sum of angles on a straight - line is $180^{\circ}$), we have the equation $(3x + 5)+(10x-7)=180$.
$$(3x + 5)+(10x-7)=180$$

Step2: Simplify the left - hand side

Combine like terms: $3x+10x+5 - 7=180$, which gives $13x-2 = 180$.
$$13x-2=180$$

Step3: Solve for x

Add 2 to both sides: $13x=180 + 2=182$. Then divide both sides by 13: $x=\frac{182}{13}=14$.
$$x = 14$$

Step4: Find the measure of each angle

For the first angle: $3x+5=3\times14 + 5=42 + 5=47^{\circ}$.
For the second angle: $10x-7=10\times14-7=140 - 7=133^{\circ}$.

Step1: Set up the equation

Since the two angles $(15x - 2)^{\circ}$ and $(7x + 4)^{\circ}$ are complementary (the sum of complementary angles is $90^{\circ}$), we have the equation $(15x-2)+(7x + 4)=90$.
$$(15x-2)+(7x + 4)=90$$

Step2: Simplify the left - hand side

Combine like terms: $15x+7x-2 + 4=90$, which gives $22x+2=90$.
$$22x+2=90$$

Step3: Solve for x

Subtract 2 from both sides: $22x=90 - 2=88$. Then divide both sides by 22: $x=\frac{88}{22}=4$.
$$x = 4$$

Step4: Find the measure of each angle

For the first angle: $15x-2=15\times4-2=60 - 2=58^{\circ}$.
For the second angle: $7x + 4=7\times4+4=28 + 4=32^{\circ}$.

Step1: Set up the equation

Since $\angle UVW$ and $\angle XYZ$ are complementary, $(x - 10)+(4x-10)=90$.
$$(x - 10)+(4x-10)=90$$

Step2: Simplify the left - hand side

Combine like terms: $x+4x-10-10=90$, which gives $5x-20=90$.
$$5x-20=90$$

Step3: Solve for x

Add 20 to both sides: $5x=90 + 20=110$. Then divide both sides by 5: $x=\frac{110}{5}=22$.
$$x = 22$$

Step4: Find the measure of each angle

For $\angle UVW$: $x - 10=22-10 = 12^{\circ}$.
For $\angle XYZ$: $4x-10=4\times22-10=88 - 10=78^{\circ}$.

Answer:

The angles are $47^{\circ}$ and $133^{\circ}$

10.