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, a on a horizontal ray, b on a ray below, c on a ray below b; m∠avc = 72°, m∠avb = 5x+5, m∠bvc = 8x+2
find x = ____
m∠avb = ____
m∠bvc = ____

Question

Accepted Answer

### Problem 1:
# Explanation:
## Step1: Set up the equation for segment addition
Since $ AC + CB = AB $, we have $ (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 = 44 $.
# Answer:
$ x = 7 $, $ AC = 44 $

### Problem 2:
# 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:
# Explanation:
## Step1: Use angle addition postulate
Since $ D $ is in the interior of $ \angle ABE $, $ m\angle ABD + m\angle DBE = m\angle ABE $.
# Answer:
$ m\angle ABD + m\angle DBE = m\angle ABE $

### Angle Problem (with $ V, A, B, C $):
# Explanation:
## Step1: Apply angle addition postulate
$ m\angle AUB + m\angle BVC = m\angle AVC $. So, $ (5x + 5) + (8x + 2) = 72 $.
## Step2: 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 $: $ 5(5) + 5 = 30^\circ $.
## Step4: Find $ m\angle BVC $
Substitute $ x = 5 $ into $ 8x + 2 $: $ 8(5) + 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:

Step1: Set up the equation for segment addition

Step2: Simplify and solve for \( x \)

Step3: Find \( AC \)

Step1: Use angle addition postulate

Answer:

Problem 2: