class work:
monday 9/8/25
1)
a ---- c ---- b
ac = 6x+2, cb = x+1, ab = 52
x= ____
ac= ____
2) if e lies between b and c
then ____
3) if d is in the interior of ∠abe
then ____
diagram: v connected to a, b, c; a on a horizontal line, b on a line from v, c on another line from v
if m∠avc = 72°
if m∠avb = 5x+5, m∠bvc = 8x+2
find x= ____
m∠avb= ____
m∠bvc= ____

Question

Accepted Answer

### Problem 1: Segment Addition
# Explanation:
## Step1: Apply Segment Addition Postulate
Since $ C $ is between $ A $ and $ B $, $ AC + CB = AB $. Given $ AC = 6x + 2 $, $ CB = x + 1 $, and $ AB = 52 $, we set up the equation:
$ (6x + 2) + (x + 1) = 52 $

## Step2: Simplify and Solve for $ x $
Combine like terms:
$ 7x + 3 = 52 $
Subtract 3 from both sides:
$ 7x = 49 $
Divide by 7:
$ x = 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 $, by the Segment Addition Postulate, $ BE + EC = BC $ (or $ B - E - C $ implies $ BE + EC = BC $).

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

### Problem 3: Angle Addition (Point in Interior)
# Explanation:
## Step1: Apply Angle Addition Postulate
Since $ D $ is in the interior of $ \angle ABE $, $ m\angle ABD + m\angle DBE = m\angle ABE $.

### Problem 4: Angle Addition with Variables
# Explanation:
## Step1: Apply Angle Addition Postulate
Given $ m\angle AVC = 72^\circ $, $ m\angle AUB = 5x + 5 $, $ m\angle BVC = 8x + 2 $, and $ \angle AUB + \angle BVC = \angle AVC $ (since $ B $ is in the interior of $ \angle AVC $):
$ (5x + 5) + (8x + 2) = 72 $

## Step2: Simplify and Solve for $ x $
Combine like terms:
$ 13x + 7 = 72 $
Subtract 7:
$ 13x = 65 $
Divide by 13:
$ x = 5 $

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

## Step4: Find $ m\angle BVC $
Substitute $ x = 5 $ into $ 8x + 2 $:
$ m\angle BVC = 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 Addition

Step1: Apply Segment Addition Postulate

Step2: Simplify and Solve for \( x \)

Step3: Find \( AC \)

Step1: Apply Angle Addition Postulate

Problem 4: Angle Addition with Variables

Answer:

Problem 2: Point Between Segments