Question

class work:
monday 9/8/25
1)
a ---- 6x+2 ---- c ---- x+1 ---- b
|<-------- 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 three rays (va, vb, vc); text: if ( mangle avc = 72^circ )
if ( mangle avb = 5x + 5 )
( mangle bvc = 8x + 2 )
find ( x = ) ____
( mangle avb = ) ____
( mangle 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 \( \angle ABE \), \( m\angle ABD + m\angle DBE = m\angle ABE \).

Answer:

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

Problem 2: Point Between Segments