Question

algebra in the figure, \\( \overrightarrow{k j} \\) and \\( \overrightarrow{k l} \\) are opposite rays. \\( \overrightarrow{k n} \\) bisects \\( \angle l k m \\).

1. if \\( m \angle l k m=7 x-5 \\) and \\( m \angle n k m=3 x+9 \\), find \\( m \angle l k m \\).
2. if \\( m \angle n k l=7 x-9 \\) and \\( m \angle j k m=x+3 \\), find \\( m \angle j k n \\).

Explanation:

Step1: Use the angle - bisector property

Since \( \overrightarrow{KN}\) bisects \( \angle LKM\), we have \(m\angle LKM = 2m\angle NKM\).
Given \(m\angle LKM=7x - 5\) and \(m\angle NKM = 3x + 9\), then \(7x-5=2(3x + 9)\).

Step2: Solve the equation for \(x\)

Expand the right - hand side: \(7x-5 = 6x+18\).
Subtract \(6x\) from both sides: \(7x-6x-5=6x - 6x+18\), so \(x-5 = 18\).
Add \(5\) to both sides: \(x=18 + 5=23\).

Step3: Find \(m\angle LKM\)

Substitute \(x = 23\) into \(m\angle LKM=7x - 5\).
\(m\angle LKM=7\times23-5=161-5 = 156\).

Answer:

\(m\angle LKM = 156^{\circ}\)