Question

the triangle.
$m\angle l=\square{}^{circ}$
$m\angle m=\square{}^{circ}$
$m\angle n=\square{}^{circ}$

Explanation:

Step1: Use the property of isosceles triangle

Since \( \triangle MLN \) is isosceles (\( MN = LN \)), \( \angle L=\angle M \). So \( 5x - 7=3x + 15 \).
Solve the equation \( 5x - 7=3x + 15 \):
Subtract \( 3x \) from both sides: \( 5x-3x - 7=3x-3x + 15 \), \( 2x-7 = 15 \).
Add \( 7 \) to both sides: \( 2x-7 + 7=15 + 7 \), \( 2x=22 \).
Divide both sides by \( 2 \): \( x = 11 \).

Step2: Calculate \( m\angle L \) and \( m\angle M \)

Substitute \( x = 11 \) into \( 3x + 15 \) (or \( 5x - 7 \)):
\( m\angle L=3x + 15=3\times11 + 15=33 + 15=48^{\circ} \), \( m\angle M=5x - 7=5\times11 - 7=55 - 7=48^{\circ} \).

Step3: Calculate \( m\angle N \)

Use the triangle - angle sum theorem (\( m\angle L+m\angle M+m\angle N = 180^{\circ} \)).
Let \( m\angle N=y \), then \( 48^{\circ}+48^{\circ}+y = 180^{\circ} \).
\( y=180-(48 + 48)=180 - 96=84^{\circ} \).

Answer:

\( m\angle L = 48^{\circ} \), \( m\angle M = 48^{\circ} \), \( m\angle N = 84^{\circ} \)