class work: monday 9/8/25
1) diagram: a---c---b, ac=6x+2, cb=x+1, ab=52 x=______ ac=______
2) if e lies between b and c then______
3) diagram: angle v with a, b, c if m∠avc = 72°, m∠aub=5x+5, m∠bvc=8x+2 find x=______, m∠aub=______, m∠bvc=______

Question

Accepted Answer

### Problem 1: Segment Lengths
# Explanation:
## Step1: Set up the equation
Since $ AC + CB = AB $, we have $ (6x + 2) + (x + 1) = 52 $.
Simplify the left side: $ 7x + 3 = 52 $.

## Step2: Solve for $ x $
Subtract 3 from both sides: $ 7x = 52 - 3 = 49 $.
Divide by 7: $ x = \frac{49}{7} = 7 $.

## Step3: Find $ AC $
Substitute $ x = 7 $ into $ AC = 6x + 2 $: $ AC = 6(7) + 2 = 42 + 2 = 44 $.

# Answer:
$ x = 7 $, $ AC = 44 $

### Problem 2: Point Between Segments
# Brief Explanations:
If $ E $ lies between $ B $ and $ C $, then by the segment addition postulate, $ BE + EC = BC $ (or $ B - E - C $ collinear with $ BE + EC = BC $).

# Answer:
$ BE + EC = BC $ (or $ B, E, C $ are collinear with $ BE + EC = BC $)

### Problem 3: Angle Addition with Interior Point
# Explanation:
## Step1: Set up the angle equation
Since $ D $ is in the interior of $ \angle ABE $, $ m\angle ABD + m\angle DBE = m\angle ABE $. (Note: The diagram's angle label might have a typo, assuming $ \angle ABE $ as the angle with interior point $ D $.)

## Step2: (If more info, but as per structure, state the postulate)
By the angle addition postulate, the sum of the two adjacent angles formed by a point in the interior equals the measure of the larger angle.

# Answer:
$ m\angle ABD + m\angle DBE = m\angle ABE $ (or similar based on correct angle labels)

### Angle Problem (with $ \angle AVC = 72^\circ $, $ \angle AUB = 5x + 5 $, $ \angle BVC = 8x + 2 $)
# Explanation:
## Step1: Set up the angle equation
Assuming $ \angle AVC = \angle AUB + \angle BVC $ (since $ B $ is in the interior of $ \angle AVC $), we have $ 5x + 5 + 8x + 2 = 72 $.

## Step2: Simplify and solve for $ x $
Combine like terms: $ 13x + 7 = 72 $.
Subtract 7: $ 13x = 72 - 7 = 65 $.
Divide by 13: $ x = \frac{65}{13} = 5 $.

## Step3: Find $ m\angle AUB $
Substitute $ x = 5 $ into $ 5x + 5 $: $ 5(5) + 5 = 25 + 5 = 30^\circ $.

## Step4: Find $ m\angle BVC $
Substitute $ x = 5 $ into $ 8x + 2 $: $ 8(5) + 2 = 40 + 2 = 42^\circ $.

# Answer:
$ x = 5 $, $ m\angle AUB = 30^\circ $, $ m\angle BVC = 42^\circ $

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QUESTION IMAGE

Question

Explanation:

Problem 1: Segment Lengths

Step1: Set up the equation

Step2: Solve for \( x \)

Step3: Find \( AC \)

Step1: Set up the angle equation

Step2: (If more info, but as per structure, state the postulate)

Answer:

Problem 2: Point Between Segments