5. it is known that ∠bam and ∠ctx are complementary. if m∠bam = 2x + 4 and m∠ctx = x + 20, then solve for the value of x algebraically.
6. for two supplementary angles, the measure of the larger angle is five times larger than the measure of the smaller angle. find the measure of the larger angle. justify how you found your answer.
reasoning
7. if ∠tap and ∠bre are supplementary and ∠bre is its own complement, find the measure of ∠tap. show how you arrived at your answer.
8. in the diagram below, ∠jhl and ∠ihk are both right angles
(a) name two angles that are both complements of ∠jhk.
(b) if m∠jhk = 40°, find the measure of both angles from (a).
(c) what must be true about the measures of two angles that are both complementary to the same angle? explain.

Question

Accepted Answer

### 5.
# Explanation:
## Step1: Recall complementary - angle property
Complementary angles add up to 90 degrees. So, $m\angle BAM + m\angle CTX=90$.
Given $m\angle BAM = 2x + 4$ and $m\angle CTX=x + 20$, we have the equation $(2x + 4)+(x + 20)=90$.
## Step2: Simplify the left - hand side of the equation
Combine like terms: $2x+x+4 + 20=90$, which simplifies to $3x+24 = 90$.
## Step3: Isolate the variable term
Subtract 24 from both sides of the equation: $3x+24−24=90−24$, resulting in $3x=66$.
## Step4: Solve for x
Divide both sides of the equation by 3: $\frac{3x}{3}=\frac{66}{3}$, so $x = 22$.
# Answer:
$x = 22$

### 6.
# Explanation:
## Step1: Let the measure of the smaller angle be $y$
Then the measure of the larger angle is $5y$.
Since the two angles are supplementary, they add up to 180 degrees. So, $y + 5y=180$.
## Step2: Combine like terms
$6y=180$.
## Step3: Solve for y
Divide both sides by 6: $\frac{6y}{6}=\frac{180}{6}$, so $y = 30$.
## Step4: Find the measure of the larger angle
The larger angle is $5y$. Substitute $y = 30$ into $5y$, we get $5\times30=150$ degrees.
# Answer:
150 degrees

### 7.
# Explanation:
## Step1: Recall the property of an angle that is its own complement
If an angle $\angle BRE$ is its own complement, then $\angle BRE+\angle BRE = 90$ (since complementary angles add up to 90 degrees). So, $2\angle BRE=90$, and $\angle BRE = 45$ degrees.
## Step2: Recall the property of supplementary angles
Since $\angle TAP$ and $\angle BRE$ are supplementary, $\angle TAP+\angle BRE=180$ degrees.
Substitute $\angle BRE = 45$ into the equation: $\angle TAP+45=180$.
## Step3: Solve for $\angle TAP$
Subtract 45 from both sides: $\angle TAP=180 - 45=135$ degrees.
# Answer:
135 degrees

### 8.
## (a)
# Explanation:
Since $\angle JHL$ and $\angle JHK$ are right - angles ($90$ degrees), and $\angle IHJ+\angle JHK=\angle JHL = 90$ and $\angle KHI+\angle IHJ=\angle JHK = 90$, the two angles that are complements of $\angle JHK$ are $\angle IHJ$ and $\angle KHI$.
# Answer:
$\angle IHJ$ and $\angle KHI$

## (b)
# Explanation:
If $m\angle JHK = 40$ degrees, and $\angle IHJ$ and $\angle KHI$ are complements of $\angle JHK$.
For $\angle IHJ$, since $\angle IHJ+\angle JHK = 90$, then $\angle IHJ=90 - 40=50$ degrees.
For $\angle KHI$, since $\angle KHI+\angle IHJ=\angle JHK$ and $\angle IHJ+\angle JHK = 90$, $\angle KHI = 50$ degrees.
# Answer:
50 degrees, 50 degrees

## (c)
# Explanation:
If two angles are both complementary to the same angle, let the angle be $A$, and the two complementary angles be $B$ and $C$.
We know that $B + A=90$ and $C + A=90$. Then $B=90 - A$ and $C=90 - A$. So, the measures of two angles that are both complementary to the same angle are equal.
# Answer:
The measures of the two angles are equal.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

5.

Step1: Recall complementary - angle property

Step2: Simplify the left - hand side of the equation

Step3: Isolate the variable term

Step4: Solve for x

Step1: Let the measure of the smaller angle be \(y\)

Step2: Combine like terms

Step3: Solve for y

Step4: Find the measure of the larger angle

Step1: Recall the property of an angle that is its own complement

Step2: Recall the property of supplementary angles

Step3: Solve for \(\angle TAP\)

Answer:

6.