5. ∠1 and ∠2 are complementary. solve for the measure of ∠1. ∠1 = 12x + 4, ∠2 = 9x + 2. a. 4, b. 8.3, c. 38, d. 52. 6. given that l || m, ∠3 = 7x + 12 and ∠6 = 10x − 3. find the measure of ∠3. a. 5, b. 5.8, c. 47, d. 52.6. (diagram: lines l, m, transversal with angles 1–4 on l, 5–8 on m) 7. solve for ∠j. (diagram: triangle jmk, ∠m = 73°, ∠k has angle x + 40, and a 120° angle. options: a. 20°, b. 107°, c. 47°, d. 67°) 8. transversal $\overline{ef}$ intersects $\overline{ab}$ and $\overline{cd}$, as shown in the diagram below. which statement could always be used to show $\overline{ab} \parallel \overline{cd}$? a. ∠1 and ∠5 are supplementary, b. ∠7 ≅ ∠8, c. ∠3 and ∠6 are supplementary, d. ∠2 ≅ ∠4. (diagram: $\overline{ab}$, $\overline{cd}$, transversal $\overline{ef}$ with angles 1–4 on $\overline{ab}$, 5–8 on $\overline{cd}$)

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Accepted Answer

### Question 5
# Explanation:
## Step1: Recall complementary angles sum to $90^\circ$
$\angle1 + \angle2 = 90^\circ$, so $(12x + 4) + (9x + 2) = 90$
## Step2: Simplify and solve for $x$
$21x + 6 = 90$ → $21x = 84$ → $x = 4$
## Step3: Substitute $x$ into $\angle1$'s formula
$\angle1 = 12(4) + 4 = 52$? Wait, no, wait: Wait, $12x + 4$ when $x = 4$ is $12*4 +4=52$? But wait, let's check again. Wait, $12x +4 +9x +2 = 21x +6 =90$ → $21x=84$ → $x=4$. Then $\angle1=12*4 +4=52$? But the options have D as 52. Wait, but the original marked option was B (8.3)? Wait, maybe I made a mistake. Wait, no—wait, complementary angles sum to 90, so $12x +4 +9x +2 = 90$ → $21x +6 =90$ → $21x=84$ → $x=4$. Then $\angle1=12*4 +4=52$, so D. But the marked answer was B. Wait, maybe the problem was supplementary? Wait, the user wrote "complementary". If it's supplementary (sum to 180), then $21x +6=180$ → $21x=174$ → $x≈8.285≈8.3$, which is option B. Oh! Maybe a typo: if $\angle1$ and $\angle2$ are supplementary (sum to 180), then:

## Step1: Supplementary angles sum to $180^\circ$
$(12x + 4) + (9x + 2) = 180$
## Step2: Simplify: $21x + 6 = 180$ → $21x = 174$ → $x = \frac{174}{21} ≈ 8.285 ≈ 8.3$
## Step3: Find $\angle1$: $12(8.3) + 4 ≈ 99.6 + 4 = 103.6$? No, wait, no—wait, if $x≈8.3$, then $\angle1=12x +4≈12*8.3 +4=99.6 +4=103.6$, which is not an option. Wait, the options are A.4, B.8.3 (x?), C.38, D.52. Wait, maybe the problem is $\angle1$ and $\angle2$ are complementary, so sum to 90. Then $12x +4 +9x +2=90$ → $21x=84$ → $x=4$. Then $\angle1=12*4 +4=52$ (D), $\angle2=9*4 +2=38$ (C). So maybe the original problem had a typo, but following the calculation, if complementary, answer is D. But the marked option is B, which is x≈8.3. So perhaps the problem was supplementary. Assuming supplementary (maybe a typo), then x≈8.3, but the question is to find $\angle1$. Wait, no—maybe I misread. Let's re-express:

Wait, the problem says "Solve for the measure of $\angle1$". Given $\angle1=12x +4$, $\angle2=9x +2$, complementary (sum 90). So:

$12x +4 +9x +2 = 90$

$21x +6 = 90$

$21x = 84$

$x = 4$

Then $\angle1 = 12*4 +4 = 52$. So answer D.

# Answer: D. 52

### Question 6
# Explanation:
## Step1: Identify angle relationship (l || m, so $\angle3$ and $\angle6$ are alternate interior angles? Wait, no—$\angle3$ and $\angle6$: since l || m, and the transversal, $\angle3$ and $\angle6$ are same-side interior angles? Wait, no, looking at the diagram: $\angle3$ and $\angle6$ are alternate interior angles? Wait, $\angle3$ and $\angle5$ are same-side, $\angle3$ and $\angle6$ are... Wait, in the diagram, l and m are parallel, transversal intersects them. $\angle3$ and $\angle6$: if $\angle3$ is on line l, below, and $\angle6$ is on line m, above, then they are alternate interior angles? Wait, no, alternate interior angles are equal. Wait, but $\angle3 = 7x +12$, $\angle6 =10x -3$. If they are alternate interior angles, then $7x +12 =10x -3$. Let's solve:

## Step1: Set angles equal (alternate interior angles)
$7x + 12 = 10x - 3$
## Step2: Solve for $x$
$12 + 3 = 10x -7x$ → $15 = 3x$ → $x = 5$
## Step3: Find $\angle3$
$\angle3 = 7(5) +12 = 35 +12 = 47$
# Answer: C. 47

### Question 7
# Explanation:
## Step1: Linear pair: $x + 40 + 120 = 180$ (since they form a straight line)
$x + 160 = 180$ → $x = 20$
## Step2: Triangle angle sum: $\angle J + 73^\circ + (x + 40)^\circ = 180^\circ$
Substitute $x = 20$: $\angle J + 73 + 60 = 180$ → $\angle J + 133 = 180$ → $\angle J = 47^\circ$
# Answer: C. $47^\circ$

### Question 8
# Explanation:
To show $AB || CD$, we use parallel line theorems. Let's analyze options:

- A: $\angle1$ and $\angle5$ supplementary: $\angle1$ and $\angle5$ are corresponding? No, $\angle1$ and $\angle5$ being supplementary doesn't imply parallel.
- B: $\angle7 ≅ \angle8$: Vertical angles, always equal, not related to parallel.
- C: $\angle3$ and $\angle6$ supplementary: $\angle3$ and $\angle6$ are same-side interior angles. If same-side interior angles are supplementary, lines are parallel. So this works.
- D: $\angle2 ≅ \angle4$: Vertical angles, always equal, not related to parallel.

# Answer: C. $\angle3$ and $\angle6$ are supplementary

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QUESTION IMAGE

Question

Explanation:

Question 5

Step1: Recall complementary angles sum to \(90^\circ\)

Step2: Simplify and solve for \(x\)

Step3: Substitute \(x\) into \(\angle1\)'s formula

Step1: Supplementary angles sum to \(180^\circ\)

Step2: Simplify: \(21x + 6 = 180\) → \(21x = 174\) → \(x = \frac{174}{21} ≈ 8.285 ≈ 8.3\)

Step1: Set angles equal (alternate interior angles)

Step2: Solve for \(x\)

Step3: Find \(\angle3\)

Step1: Linear pair: \(x + 40 + 120 = 180\) (since they form a straight line)

Step2: Triangle angle sum: \(\angle J + 73^\circ + (x + 40)^\circ = 180^\circ\)

Answer:

Question 6