directions: find the value of x so that l || m. state the converse used
6. diagram with lines l, m, angle 53°, (8x - 9)°
converse: ______
7. diagram with lines l, m, angle (2x - 4)°, (5x + 20)°
converse: ______
8. diagram with lines l, m, angles (6x + 7)°, (4x - 3)°
converse: ______
9. diagram with lines l, m, angles (5x - 22)°, (3x + 32)°
converse: ______
10. diagram with lines l, m, angles (4x - 13)°, (2x + 37)°
converse: ______
11. diagram with lines l, m, angles (26x + 9)°, (7x - 15)°
converse: ______
12. diagram with angles 87°, 103°, (10x + 17)°
converse: ______
13. diagram with angles (14x)°, 48°, (28x)°
converse: ______

Question

Accepted Answer

Let's solve problem 6 first (since it has some initial work). The goal is to find $ x $ so that $ l \parallel m $, using the appropriate converse (like consecutive interior angles, alternate interior, etc.).

### Problem 6:
We have a transversal cutting $ l $ and $ m $. One angle on $ l $ is $ 55^\circ $, and the angle on $ m $ is $ (8x - 9)^\circ $. Wait, maybe they are consecutive interior angles? Wait, if $ l \parallel m $, consecutive interior angles are supplementary (sum to $ 180^\circ $)? Wait, no, maybe alternate interior or same - side? Wait, the initial work has $ 53 + 8x - 9 = 180 $? Wait, maybe a typo. Wait, $ 180 - 55=125 $? No, wait, maybe the angle adjacent to $ 55^\circ $ is $ 180 - 55 = 125^\circ $, and then $ 8x - 9 = 125 $? Wait, let's re - examine.

#### Step 1: Identify the angle relationship
If $ l\parallel m $, and the angles are same - side interior angles (consecutive interior angles), they are supplementary. So the angle on $ l $ is $ 55^\circ $, so the adjacent angle (same - side interior with the angle $ (8x - 9)^\circ $ on $ m $) is $ 180 - 55=125^\circ $. So we set $ 8x - 9 = 125 $.

#### Step 2: Solve for $ x $
Add 9 to both sides: $ 8x=125 + 9=134 $
Divide both sides by 8: $ x=\frac{134}{8} = 16.75 $? Wait, no, maybe the angle on $ l $ is $ 53^\circ $ (a typo in the problem, since the initial work has 53). Let's assume the angle on $ l $ is $ 53^\circ $. Then same - side interior angles: $ 53+(8x - 9)=180 $
Simplify: $ 8x+44 = 180 $
Subtract 44: $ 8x=180 - 44 = 136 $
Divide by 8: $ x = 17 $

### Problem 7:
Angles are $ (2x - 4)^\circ $ and $ (3x + 20)^\circ $. If $ l\parallel m $, these could be alternate interior angles (if the transversal is such that they are alternate). Wait, alternate interior angles are equal. So $ 2x-4=3x + 20 $? No, that would give negative $ x $. Wait, maybe same - side exterior? No, maybe consecutive interior? Wait, if they are same - side interior, $ (2x - 4)+(3x + 20)=180 $
Simplify: $ 5x+16 = 180 $
$ 5x=164 $, $ x = 32.8 $. No, maybe alternate exterior? Wait, maybe the angles are corresponding angles. If they are corresponding, then $ 2x-4 = 3x + 20 $ is wrong. Wait, maybe the angle $ (3x + 20)^\circ $ is the supplement of $ (2x - 4)^\circ $ if they are same - side. Wait, let's check the diagram (mentally). If $ l\parallel m $, and the transversal cuts them, and the two angles are same - side interior, then $ (2x - 4)+(3x + 20)=180 $
$ 5x+16 = 180 $
$ 5x=164 $
$ x = 32.8 $. But maybe I made a mistake. Alternatively, if they are alternate interior, $ 2x-4=3x + 20 $ gives $ x=-24 $, which is impossible. So maybe same - side exterior? Same - side exterior angles are also supplementary. So $ (2x - 4)+(3x + 20)=180 $, same as above. So $ x = 32.8 $. But this seems odd. Maybe the angles are corresponding. Wait, maybe the diagram is different. Let's assume that the two angles are alternate interior angles. Wait, no, let's re - express.

Wait, maybe the angle $ (3x + 20)^\circ $ is vertical to an angle that is same - side with $ (2x - 4)^\circ $. This is getting confusing. Let's move to problem 8.

### Problem 8:
Angles are $ (6x + 7)^\circ $ and $ (4x - 3)^\circ $. If $ l\parallel m $, and these are same - side interior angles (since the lines are cut by a transversal and the angles are on the same side), then $ (6x + 7)+(4x - 3)=180 $
Simplify: $ 10x + 4=180 $
$ 10x=176 $
$ x = 17.6 $. Or if they are alternate interior, $ 6x + 7=4x - 3 $, $ 2x=-10 $, $ x=-5 $ (impossible). So same - side interior, $ x = 17.6 $.

### Problem 9:
Angles are $ (5x - 22)^\circ $ and $ (3x + 5)^\circ $. If $ l\parallel m $, and these are alternate interior angles, then $ 5x-22=3x + 5 $
$ 2x=27 $
$ x = 13.5 $. Or if same - side, $ (5x - 22)+(3x + 5)=180 $, $ 8x-17 = 180 $, $ 8x=197 $, $ x = 24.625 $. Let's assume alternate interior (since the angles are on alternate sides of the transversal). So $ x = 13.5 $.

### Problem 10:
Angles are $ (4x - 13)^\circ $ and $ (2x + 37)^\circ $. If $ l\parallel m $, and these are corresponding angles (since the lines are parallel and cut by a transversal), then $ 4x-13=2x + 37 $
$ 2x=50 $
$ x = 25 $. Or if same - side, $ (4x - 13)+(2x + 37)=180 $, $ 6x + 24=180 $, $ 6x=156 $, $ x = 26 $. Let's check the diagram (mentally). If the angles are corresponding, $ x = 25 $.

### Problem 11:
Angles are $ (26x + 9)^\circ $ and $ (7x - 15)^\circ $. If $ l\parallel m $, and these are alternate interior angles, then $ 26x+9=7x - 15 $, $ 19x=-24 $, $ x$ negative. So same - side interior: $ (26x + 9)+(7x - 15)=180 $
$ 33x-6 = 180 $
$ 33x=186 $
$ x=\frac{186}{33}=\frac{62}{11}\approx5.64 $. Or alternate exterior? Same as same - side interior.

### Problem 12:
Angles: one is $ 87^\circ $, another is $ 15^\circ $, and $ (10x + 17)^\circ $. Maybe the sum of $ 87^\circ $ and $ 15^\circ $ is related to $ (10x + 17)^\circ $ if $ l\parallel m $ (using triangle or exterior angle). Wait, if $ l\parallel m $, the exterior angle of the triangle formed by the transversal and the two lines would be equal to the sum of the two remote interior angles. So $ 10x + 17=87 + 15 $
$ 10x+17 = 102 $
$ 10x=85 $
$ x = 8.5 $

### Problem 13:
Angles: $ 125x^\circ $, $ 14x^\circ $, and $ 48^\circ $. If $ l\parallel m $, the sum of the angles on one side (since it's a triangle and the lines are parallel) should satisfy the parallel line angle relationships. Maybe $ 125x+14x + 48=180 $ (if it's a triangle with a transversal)
$ 139x+48 = 180 $
$ 139x=132 $
$ x=\frac{132}{139}\approx0.95 $. Or maybe $ 125x=14x + 48 $ (alternate interior), $ 111x=48 $, $ x=\frac{48}{111}=\frac{16}{37}\approx0.43 $.

Since the user didn't specify which problem, let's take problem 6 (with the initial work correction):

# Explanation:
## Step 1: Identify angle relationship
If $ l\parallel m $, same - side interior angles are supplementary. The angle on $ l $ is $ 53^\circ $ (assuming a typo), so $ 53+(8x - 9)=180 $.
## Step 2: Simplify the equation
$ 8x + 44=180 $
## Step 3: Solve for $ x $
Subtract 44: $ 8x=136 $
Divide by 8: $ x = 17 $

# Answer:
$ x = 17 $ (using the converse of consecutive interior angles theorem)

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QUESTION IMAGE

Question

Explanation:

Problem 6:

Step 1: Identify the angle relationship

Step 2: Solve for \( x \)

Problem 7:

Problem 8:

Step 1: Identify angle relationship

Step 2: Simplify the equation

Step 3: Solve for \( x \)

Answer: