Question

1. if \\( \overline{lk} \cong \overline{mk} \\), \\( lk = 7x - 10 \\), \\( kn = x + 3 \\), \\( mn = 9x - 11 \\), and \\( kj = 28 \\), find \\( lj \\).

Explanation:

Step1: Analyze MN's composition

Since \( MN = MK + KN \) and \( LK\cong MK \) (so \( MK = LK = 7x - 10 \)), substitute \( MK \) and \( KN \) into \( MN \)'s formula.
\( MN=(7x - 10)+(x + 3) \)
Simplify: \( MN = 8x - 7 \)

Step2: Solve for x using MN's equation

We know \( MN = 9x - 11 \), so set \( 8x - 7 = 9x - 11 \)
Subtract \( 8x \) from both sides: \( - 7=x - 11 \)
Add 11 to both sides: \( x = 4 \)

Step3: Find LK's length

Substitute \( x = 4 \) into \( LK = 7x - 10 \)
\( LK = 7(4)-10=28 - 10 = 18 \)

Step4: Find LJ's length

\( LJ = LK + KJ \), we know \( LK = 18 \) and \( KJ = 28 \)
\( LJ=18 + 28 = 46 \)

Answer:

\( 46 \)