Question

class work: monday 9/8/25

1. diagram: a---c---b, ac=6x+2, cb=x+1, ab=52 x=__, ac=__
2. if e lies between b and c then ____
3. diagram: angle at v, a, b, c on rays if m∠auc=72°, m∠aub=5x+5, m∠bvc=8x+2, find x=__, m∠aub=, 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: Use angle addition postulate

Given \( m\angle AVC = 72^\circ \), \( m\angle AUB = 5x + 5 \), \( m\angle BVC = 8x + 2 \), and \( \angle AVC = \angle AUB + \angle BVC \), 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 \( 5x + 5 \): \( 5(5) + 5 = 25 + 5 = 30^\circ \).

Step4: Find \( m\angle BVC \)

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

Answer:

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

Problem 2: