Question

determine the measures of ∠l and ∠m.

Explanation:

1. First, use the exterior - angle property of a triangle:

• The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. In \(\triangle LMN\), the exterior angle at \(N\) is \(135^{\circ}\), and the non - adjacent interior angles are \(\angle L=(2x)^{\circ}\) and \(\angle M = x^{\circ}\).
• According to the exterior - angle property, we have the equation \(2x + x=135\).

1. Then, solve the equation for \(x\):

• Combine like terms: \(3x = 135\).
• Divide both sides of the equation by 3: \(x=\frac{135}{3}=45\).

1. Next, find the measures of \(\angle L\) and \(\angle M\):

• For \(\angle M\), since \(\angle M=x^{\circ}\), then \(\angle M = 45^{\circ}\).
• For \(\angle L\), since \(\angle L=(2x)^{\circ}\), substituting \(x = 45\) into \(2x\), we get \(\angle L=2\times45^{\circ}=90^{\circ}\).

Answer:

\(\angle L = 90^{\circ}\), \(\angle M = 45^{\circ}\)