Question
Triangle LMN is isosceles with <M ~= <L = 36. What are the measures of angles M and N?
Answers
Answered by
GPT-4o mini
In an isosceles triangle, two angles are equal. Given that triangle LMN is isosceles with \( \angle M \approx \angle L = 36^\circ \), it follows that:
\[
\angle M = 36^\circ
\]
\[
\angle L = 36^\circ
\]
To find the measure of angle \( N \), we can use the fact that the sum of the angles in any triangle is \( 180^\circ \). Therefore, we have:
\[
\angle M + \angle L + \angle N = 180^\circ
\]
Substituting the known values:
\[
36^\circ + 36^\circ + \angle N = 180^\circ
\]
This simplifies to:
\[
72^\circ + \angle N = 180^\circ
\]
Now, we can solve for \( \angle N \):
\[
\angle N = 180^\circ - 72^\circ = 108^\circ
\]
Thus, the measures of the angles in triangle LMN are:
\[
\angle M = 36^\circ, \quad \angle L = 36^\circ, \quad \angle N = 108^\circ
\]
\[
\angle M = 36^\circ
\]
\[
\angle L = 36^\circ
\]
To find the measure of angle \( N \), we can use the fact that the sum of the angles in any triangle is \( 180^\circ \). Therefore, we have:
\[
\angle M + \angle L + \angle N = 180^\circ
\]
Substituting the known values:
\[
36^\circ + 36^\circ + \angle N = 180^\circ
\]
This simplifies to:
\[
72^\circ + \angle N = 180^\circ
\]
Now, we can solve for \( \angle N \):
\[
\angle N = 180^\circ - 72^\circ = 108^\circ
\]
Thus, the measures of the angles in triangle LMN are:
\[
\angle M = 36^\circ, \quad \angle L = 36^\circ, \quad \angle N = 108^\circ
\]