5. it is known that ∠ham and ∠ctx are complementary. if m∠ham = 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, ∠jhi and ∠ihk are both right angles.
(a) name two angles that are both complements of ∠ihk.
(b) if m∠ihk = 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 sum to 90°. So, $m\angle HAM + m\angle CTX=90^{\circ}$.
## Step2: Substitute angle - measures
Substitute $m\angle HAM = 2x + 4$ and $m\angle CTX=x + 20$ into the equation: $(2x + 4)+(x + 20)=90$.
## Step3: Simplify the left - hand side
Combine like terms: $2x+x+4 + 20=90$, which gives $3x+24 = 90$.
## Step4: Isolate the variable term
Subtract 24 from both sides: $3x=90 - 24$, so $3x=66$.
## Step5: Solve for x
Divide both sides by 3: $x=\frac{66}{3}=22$.
# Answer:
$x = 22$

### 6.
# Explanation:
## Step1: Let the smaller angle be $x$
Let the measure of the smaller angle be $x$. Then the measure of the larger angle is $5x$.
## Step2: Recall supplementary - angle property
Supplementary angles sum to 180°. So, $x + 5x=180$.
## Step3: Combine like terms
$6x=180$.
## Step4: Solve for x
Divide both sides by 6: $x=\frac{180}{6}=30$.
## Step5: Find the larger angle
The larger angle is $5x$. Substitute $x = 30$ into $5x$, so $5x=5\times30 = 150^{\circ}$.
# Answer:
The measure of the larger angle is $150^{\circ}$

### 7.
# Explanation:
## Step1: Let $m\angle BRE=x$
If an angle is its own complement, then $x+x = 90^{\circ}$ (since complementary angles sum to 90°).
## Step2: Solve for $x$
Combining like terms gives $2x=90^{\circ}$, so $x = 45^{\circ}$.
## Step3: Use supplementary - angle property
Since $\angle TAP$ and $\angle BRE$ are supplementary ($\angle TAP+\angle BRE = 180^{\circ}$), and $m\angle BRE = 45^{\circ}$, then $m\angle TAP=180 - 45=135^{\circ}$.
# Answer:
$m\angle TAP = 135^{\circ}$

### 8.
(a)
# Explanation:
Since $\angle JHI$ and $\angle IHK$ are right - angles ($90^{\circ}$), and complementary angles sum to 90°. $\angle JHI-\angle IHK=\angle JHK$ and $\angle IHL$ are complements of $\angle IHK$ because $\angle IHK+\angle JHK = 90^{\circ}$ and $\angle IHK+\angle IHL=90^{\circ}$ (assuming $\angle JHI = 90^{\circ}$ and the angles are formed as in the diagram).
# Answer:
$\angle JHK$ and $\angle IHL$

(b)
# Explanation:
If $m\angle IHK = 40^{\circ}$, and $\angle JHK$ is complementary to $\angle IHK$, then $m\angle JHK=90 - 40=50^{\circ}$. Also, $\angle IHL$ is complementary to $\angle IHK$, so $m\angle IHL = 50^{\circ}$ (because angles complementary to the same angle are congruent).
# Answer:
$m\angle JHK = 50^{\circ}$, $m\angle IHL = 50^{\circ}$

(c)
# Explanation:
If two angles are complementary to the same angle, let the angle be $A$, and the two complementary angles be $B$ and $C$. Then $B + A=90^{\circ}$ and $C + A=90^{\circ}$. Subtracting the first equation from the second gives $C - B=0$, so $B = C$.
# Answer:
The measures of two angles that are both complementary to the same angle are equal.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

5.

Step1: Recall complementary - angle property

Step2: Substitute angle - measures

Step3: Simplify the left - hand side

Step4: Isolate the variable term

Step5: Solve for x

Step1: Let the smaller angle be \(x\)

Step2: Recall supplementary - angle property

Step3: Combine like terms

Step4: Solve for x

Step5: Find the larger angle

Step1: Let \(m\angle BRE=x\)

Step2: Solve for \(x\)

Step3: Use supplementary - angle property

Answer:

6.