Question

class work: monday 9/8/25

1. diagram: a---c---b, ac labeled 6x+2, cb labeled x+1, ab total 52 x=___ ac=___
2. if e lies between b and c then_____
3. if d is in the interior of ∠abe then_____

diagram: v with rays to a, b, c; m∠avc=72°, m∠aub=5x+5, m∠bvc=8x+2 find x=___, m∠aub=_, m∠bvc=___

Explanation:

Problem 1: Segment Addition

Step1: Apply Segment Addition Postulate

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

Step2: Simplify and Solve for \( x \)

Combine like terms: \( 7x + 3 = 52 \). Subtract 3: \( 7x = 49 \). Divide by 7: \( x = 7 \).

Step3: Find \( AC \)

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

If \( E \) lies between \( B \) and \( C \), by the Segment Addition Postulate, \( BE + EC = BC \) (or \( BB + BE = BC \) depending on notation, but generally \( BE + EC = BC \)).

Step1: Apply Angle Addition Postulate

Since \( D \) is in the interior of \( \angle ABE \), \( m\angle ABD + m\angle DBE = m\angle ABE \). (Note: Exact angles depend on diagram, but the postulate is \( \text{Sum of adjacent angles} = \text{Total angle} \))

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: Point Between Segments