Two straight lines intersect at point A. On one of the lines, points B and C are taken

Two straight lines intersect at point A. On one of the lines, points B and C are taken, and on the other – P and K so that AB = AC, AP = AK. Prove that BP = CK.

Let us prove that triangles BAP and CAK are equal.

The angle BAP is equal to the angle CAK as the vertical angles at the intersection of straight lines BC and KP.

By condition, AB = AC, AK = AP, then the triangles BAP and CAK are equal on two sides and the angle between them.

Since the triangle BAP is equal to the triangle CAK, then BP = CK, as required.