In an isosceles triangle ABC, straight lines are drawn from the ends of the base AC, which make equal angles

In an isosceles triangle ABC, straight lines are drawn from the ends of the base AC, which make equal angles with the base and intersect at point K. Prove that triangles ABK and BCK are equal.

1) Let’s make a diagram:

Given:

ABC – isosceles;

angle СAK = angle AСK;

Prove:

AВK = ВСK;

Evidence:

1) Consider the AKC:

Since the angle СAK = angle AСK, then the triangle AKС is isosceles and AK = KС;

2) Consider AВK and ВСK:

AB = BC (because ABC is isosceles);

AK = KС;

ВK – general;

Therefore, AВK = ВСK on 3 grounds (on three sides);

Q.E.D.