Question

1. in \\( \triangle k l m \\), if \\( \angle k \cong \angle l, k l = 9 x - 40, l m = 7 x - 37 \\), and \\( k m = 3 x + 23 \\), find the value of \\( x \\) and the measure of each side.
2. in \\( \triangle x y z \\), if \\( \angle x \cong \angle z, x y = 13 x - 21, y z = 8 x - 6 \\), and \\( x z = x + 4 \\), find the value of \\( x \\) and the

Explanation:

Step1: Use the property of isosceles triangle

Since \(\angle K\cong\angle L\) in \(\triangle KLM\), then \(KM = LM\) (sides opposite to congruent angles are equal).
Set up the equation \(3x + 23=7x-37\).

Step2: Solve the equation for \(x\)

Subtract \(3x\) from both sides: \(23 = 7x-3x-37\), so \(23=4x - 37\).
Add \(37\) to both sides: \(23 + 37=4x\), i.e., \(60 = 4x\).
Divide both sides by \(4\): \(x=\frac{60}{4}=15\).

Step3: Find the length of \(KL\)

Substitute \(x = 15\) into \(KL=9x - 40\).
\(KL=9\times15-40=135 - 40=95\).

Step4: Find the length of \(LM\)

Substitute \(x = 15\) into \(LM=7x-37\).
\(LM=7\times15-37=105 - 37 = 68\).

Step5: Find the length of \(KM\)

Substitute \(x = 15\) into \(KM=3x + 23\).
\(KM=3\times15+23=45+23=68\).

Answer:

\(x = 15\), \(KL = 95\), \(LM = 68\), \(KM = 68\)