Question

1. in triangle lmn, the exterior angle at vertex n is 105°. if the ratio of the measures of ∠l to ∠m is 2:3, find the measure of ∠m.

Explanation:

Step1: Recall exterior - angle property

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. Let $\angle L = 2x$ and $\angle M=3x$.

Step2: Apply the exterior - angle formula

We know that the exterior angle at vertex $N$ is $105^{\circ}$, and by the exterior - angle property of a triangle, $\angle L+\angle M = 105^{\circ}$.

Step3: Substitute the values of $\angle L$ and $\angle M$

Substituting $\angle L = 2x$ and $\angle M = 3x$ into $\angle L+\angle M = 105^{\circ}$, we get $2x + 3x=105^{\circ}$, which simplifies to $5x = 105^{\circ}$.

Step4: Solve for $x$

Dividing both sides of the equation $5x = 105^{\circ}$ by 5, we have $x=\frac{105^{\circ}}{5}=21^{\circ}$.

Step5: Find the measure of $\angle M$

Since $\angle M = 3x$, substituting $x = 21^{\circ}$, we get $\angle M=3\times21^{\circ}=63^{\circ}$.

Answer:

$63^{\circ}$