In a triangle KLM KL = 2, angle K = 60, angle L = 70. Point D was marked on the KM side so that KD = 1.

In a triangle KLM KL = 2, angle K = 60, angle L = 70. Point D was marked on the KM side so that KD = 1. Find the corners of the LDM triangle.

The sum of the angles of any triangle is 180 degrees, which means angle M = 180 – angle K – angle L = 180 – 60 – 70 = 50 degrees.
Consider the triangle KLD. According to the cosine theorem, the square of the side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of these sides by the cosine of the angle between them. Find LD:
LD² = KD² + LK² – 2 * KD * LK * cosK = 1² + 2² – 2 * 1 * 2 * cos60 = 5 – 4 * 0.5 = 5 – 2 = 3;
LD = √3.
According to the sine theorem, the sides of a triangle are related as the sines of their opposite angles.
LD / sin K = KL / sin LDK;
sin LDK = KL * sin K / LD = 2 * sin60 / √3 = 1;
angle LDK = arcsin (1) = 90 degrees.
The angles LDK and LDM are adjacent, so their sum is 180 degrees. Hence, angle LDM = 180 – angle LDK = 180 – 90 = 90 degrees.
angle MLD = 180 – angle M – angle LDM = 180 – 50 – 90 = 40 degrees.
The angles of the LDM triangle are 50, 90 and 40 degrees.