Question

class work:
monday 9/8/25
1)
a ---- c ---- b
6x+2 x+1
|--------52--------|
x= ____
ac= ____

1. if e lies between b and c

then ____

1. if d is in the interior of ∠abe

then ____

diagram: v with points a, b, c on rays; text: if m∠avc = 72°
if m∠aub = 5x + 5 (likely typo, intended ∠avb)
m∠bvc = 8x + 2
find x = ____
m∠avb = ____
m∠bvc = ____

Explanation:

Problem 1:

Step1: Use segment addition postulate

Since \( C \) is between \( A \) and \( B \), \( AC + CB = AB \). So, \( (6x + 2) + (x + 1) = 52 \).

Step2: Simplify the equation

Combine like terms: \( 7x + 3 = 52 \).

Step3: Solve for \( x \)

Subtract 3 from both sides: \( 7x = 52 - 3 = 49 \). Then divide by 7: \( x = \frac{49}{7} = 7 \).

Step4: Find \( AC \)

Substitute \( x = 7 \) into \( AC = 6x + 2 \): \( AC = 6(7) + 2 = 42 + 2 = 44 \).

If \( E \) lies between \( B \) and \( C \), by the segment addition postulate, \( BE + EC = BC \) (or \( BB + BE = BC \) depending on notation, but generally the sum of the two smaller segments equals the larger segment).

Step1: Use angle addition postulate

Since \( D \) is in the interior of \( \angle ABE \), \( m\angle ABD + m\angle DBE = m\angle ABE \). (Note: The exact angles depend on the diagram, but the general principle is the sum of the two smaller angles equals the larger angle.)

Step2: (For the angle problem with \( V, A, B, C \))

Step1: Use angle addition postulate

\( m\angle AUB + m\angle BVC = m\angle AVC \). So, \( (5x + 5) + (8x + 2) = 72 \).

Step2: Simplify the equation

Combine like terms: \( 13x + 7 = 72 \).

Step3: Solve for \( x \)

Subtract 7: \( 13x = 72 - 7 = 65 \). Divide by 13: \( x = \frac{65}{13} = 5 \).

Step4: Find \( m\angle AUB \)

Substitute \( x = 5 \): \( 5(5) + 5 = 25 + 5 = 30^\circ \).

Step5: Find \( m\angle BVC \)

Substitute \( x = 5 \): \( 8(5) + 2 = 40 + 2 = 42^\circ \).

Answer:

\( x = 7 \), \( AC = 44 \)

Problem 2: