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 connected to a, b, c (a on horizontal line, b on middle ray, c on lower ray)
if m∠avc = 72°
if m∠avb = 5x+5
m∠bvc = 8x+2
find x= ____
m∠avb= ____
m∠bvc= ____

Explanation:

Problem 1:

Step1: Set up the equation (AC + CB = AB)

Since \( AC = 6x + 2 \), \( CB = x + 1 \), and \( AB = 52 \), 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 \( AC = 6x + 2 \): \( AC = 6(7) + 2 = 42 + 2 = 44 \).

If \( E \) lies between \( B \) and \( C \), then by the segment addition postulate, \( BE + EC = BC \) (or \( B - E - C \) collinear with \( BE + EC = BC \)).

Step1: Set up the angle addition equation

Since \( D \) is in the interior of \( \angle ABE \), \( m\angle ABD + m\angle DBE = m\angle ABE \). (But the diagram might have specific labels; assuming standard angle addition, the relationship is the sum of the two smaller angles equals the larger angle.)

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

Given \( m\angle AVC = 72^\circ \), \( m\angle AVB = 5x + 5 \), \( m\angle BVC = 8x + 2 \). By angle addition, \( m\angle AVB + m\angle BVC = m\angle AVC \). So \( (5x + 5) + (8x + 2) = 72 \).

Step3: Solve for \( x \)

Combine like terms: \( 13x + 7 = 72 \). Subtract 7: \( 13x = 65 \). Divide by 13: \( x = 5 \).

Step4: Find \( m\angle AVB \) and \( m\angle BVC \)

\( m\angle AVB = 5(5) + 5 = 30^\circ \). \( m\angle BVC = 8(5) + 2 = 42^\circ \).

Answer:

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

Problem 2: