A straight line crossing the base of an isosceles triangle is parallel to one of the lateral sides.

A straight line crossing the base of an isosceles triangle is parallel to one of the lateral sides. Prove that the triangle is isosceles.

1) Let the triangle ABC be isosceles, AB = BC, <A = <C, as the angles at the base.

2) Let’s draw a straight line В1С1 parallel to the side of the BC.

3) Consider the triangle AB1C1, which is similar to the triangle ABC, since the following angles were formed in the new triangle.

<AC1B1 = <ACB; <AB1C1 = <ABC, as the corresponding angles at parallel straight lines BC and B1C1.

4) But similar triangles have the same appearance, that is, they are also isosceles. You can also prove that B1C1: BC = AB1: AB; B1C1: AB1 = BC: AB = 1; therefore, B1C1 = AB1, the triangle is isosceles.