Question

class work: monday 9/8/25 1)
a---c---b (ac=6x+2, cb=x+1, ab=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 below, c below b
if m∠avc = 72°
if 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 \( B - E - C \) collinear with \( BE + EC = BC \)).

Step1: Apply Angle Addition Postulate

Since \( B \) is in the interior of \( \angle AVC \), \( m\angle AUB + m\angle BVC = m\angle AVC \). So, \( (5x + 5) + (8x + 2) = 72 \).

Step2: Simplify and 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:

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

Problem 2: Point Between Segments