13. if (mangle deg=(5x - 4)^{circ}), (mangle gef=(7x - 8)^{circ}), (mangle deh=(9y + 5)^{circ}), find the values of (x) and (y).
14. (angle r) and (angle s) are complementary angles. if (mangle r=(12x - 3)^{circ}) and (mangle s=(7x - 2)^{circ}), find (mangle r).
15. (angle p) and (angle q) are supplementary angles. if (mangle p=(4x + 1)^{circ}) and (mangle q=(9x - 3)^{circ}), find (mangle q).
16. (angle1) and (angle2) form a linear - pair. the measure of (angle2) is six more than twice the measure of (angle1). find (mangle2).
17. (angle j) and (angle k) are complementary angles. the measure of (angle j) is 18 less than the measure of (angle k). find the measure of each angle.

Question

Accepted Answer

### 13.
# Explanation:
## Step1: Set up equation for $x$
Since $\angle DEG$ and $\angle GEF$ are adjacent - angles and $\angle DEH$ is a straight - angle ($180^{\circ}$), and $\angle DEG+\angle GEF = \angle DEH$, we have $(5x - 4)+(7x - 8)=180$.
First, combine like terms:
$$
\begin{align*}
5x-4 + 7x-8&=180\
12x-12&=180
\end{align*}
$$
## Step2: Solve for $x$
Add 12 to both sides of the equation: $12x=180 + 12=192$. Then divide both sides by 12: $x=\frac{192}{12}=16$.
## Step3: Solve for $y$
Since $\angle DEH = 180^{\circ}$, we set $9y + 5=180$. Subtract 5 from both sides: $9y=180 - 5 = 175$. Then $y=\frac{175}{9}$.
# Answer:
$x = 16,y=\frac{175}{9}$

### 14.
# Explanation:
## Step1: Use the complementary - angle property
Since $\angle R$ and $\angle S$ are complementary angles, $m\angle R+m\angle S = 90^{\circ}$. So, $(12x - 3)+(7x - 2)=90$.
Combine like terms: $12x+7x-3 - 2=90$, which simplifies to $19x-5 = 90$.
## Step2: Solve for $x$
Add 5 to both sides: $19x=90 + 5=95$. Divide both sides by 19: $x = 5$.
## Step3: Find $m\angle R$
Substitute $x = 5$ into the expression for $m\angle R$: $m\angle R=12x-3=12\times5-3=60 - 3=57^{\circ}$.
# Answer:
$m\angle R = 57^{\circ}$

### 15.
# Explanation:
## Step1: Use the supplementary - angle property
Since $\angle P$ and $\angle Q$ are supplementary angles, $m\angle P+m\angle Q = 180^{\circ}$. So, $(4x + 1)+(9x - 3)=180$.
Combine like terms: $4x+9x+1 - 3=180$, which simplifies to $13x-2 = 180$.
## Step2: Solve for $x$
Add 2 to both sides: $13x=180 + 2=182$. Divide both sides by 13: $x = 14$.
## Step3: Find $m\angle Q$
Substitute $x = 14$ into the expression for $m\angle Q$: $m\angle Q=9x-3=9\times14-3=126 - 3=123^{\circ}$.
# Answer:
$m\angle Q = 123^{\circ}$

### 16.
# Explanation:
## Step1: Set up equations based on the linear - pair property
Let $m\angle1=x$. Then $m\angle2=2x + 6$. Since $\angle1$ and $\angle2$ form a linear pair, $m\angle1+m\angle2 = 180^{\circ}$. So, $x+(2x + 6)=180$.
Combine like terms: $x+2x+6=180$, which simplifies to $3x+6 = 180$.
## Step2: Solve for $x$
Subtract 6 from both sides: $3x=180 - 6=174$. Divide both sides by 3: $x = 58$.
## Step3: Find $m\angle2$
Substitute $x = 58$ into the expression for $m\angle2$: $m\angle2=2x + 6=2\times58+6=116 + 6=122^{\circ}$.
# Answer:
$m\angle2 = 122^{\circ}$

### 17.
# Explanation:
## Step1: Set up equations based on the complementary - angle property
Let $m\angle K=x$. Then $m\angle J=x - 18$. Since $\angle J$ and $\angle K$ are complementary angles, $m\angle J+m\angle K = 90^{\circ}$. So, $(x - 18)+x=90$.
Combine like terms: $x+x-18=90$, which simplifies to $2x-18 = 90$.
## Step2: Solve for $x$
Add 18 to both sides: $2x=90 + 18=108$. Divide both sides by 2: $x = 54$.
## Step3: Find the measures of the angles
$m\angle K=x = 54^{\circ}$, and $m\angle J=x - 18=54 - 18=36^{\circ}$.
# Answer:
$m\angle J = 36^{\circ},m\angle K = 54^{\circ}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

13.

Step1: Set up equation for \(x\)

Step2: Solve for \(x\)

Step3: Solve for \(y\)

Step1: Use the complementary - angle property

Step2: Solve for \(x\)

Step3: Find \(m\angle R\)

Step1: Use the supplementary - angle property

Step2: Solve for \(x\)

Step3: Find \(m\angle Q\)

Answer:

14.