In the rectangle ABCD, the bisector of angle A is drawn which intersects the side BC at point O.

In the rectangle ABCD, the bisector of angle A is drawn which intersects the side BC at point O. Prove that the triangle ABO isosceles.

After completing the drawing for the task, we write down the condition and requirement of the task.

Given: ABCD is a rectangle, AO is a bisector, AO intersects BC at point O.

Prove that the ABO triangle is isosceles.

Proof:

Consider the ABO triangle:

The ABO triangle is rectangular, since the corner B in the rectangle is straight.
The AAO angle is 90 °: 2 = 45 °, since AO is the bisector of the right angle A of this rectangle.
The BOA angle is 90 ° – 45 ° = 45 ° according to the property of the angles of a right-angled triangle.
The angles of the BAO and BOA are 45 degrees each, which means they are equal.
Equal sides lie opposite equal angles in a triangle. AB = BO.
The ABO triangle is isosceles.