The bisector MS is drawn in an isosceles triangle KLM with base KM. Find the angles of the triangle KLM

The bisector MS is drawn in an isosceles triangle KLM with base KM. Find the angles of the triangle KLM if the MSK angle is 105 degrees.

Since MS is a bisector, it divides the angle M in half, which means ∠SMK = 0.5 * ∠M = 0.5 * ∠K, since the angles M and K are equal as the angles at the base of the CM of an isosceles triangle.

Consider triangle SMK. By condition, ∠MSK = 105 °, the sum of the angles of the triangle is 180 °, which means:

∠К + ∠SMK = 180 ° – ∠MSK = 180 ° – 105 ° = 75 °;

∠K + 0.5 * ∠K = 1.5 * ∠K = 75 °;

∠K = 75 ° / 1.5 = 50 °.

Consequently, the angles M and K at the base of the CM are equal to 50 °.

∠K = ∠М = 50 °.

Angle L at the vertex of this triangle:

∠L = 180 ° – ∠K – ∠M = 180 ° – 50 ° – 50 ° = 80 °.