Refer to the attached image.
Given : Triangle ABC is isosceles, measure of ∠B equal to 36° and AD is an angle bisector.
To prove: Triangle CDA and ADB are isosceles
Proof:
Since triangle ABC is isosceles,
therefore AB=BC
Now,
(Angles opposite to the equal opposite sides are always equal)
Let
Therefore, by angle sum property which states
"The sum of all the angles of a triangle is 180 degrees"
Hence,
Since, AD is an angle bisector.
Therefore, it divides angle A into two equal parts.
Therefore,
Now, consider triangle ABD,
here since
Therefore, AD = BD
"By the converse of the base angles theorem, which states that if two angles of a triangle are congruent, then sides opposite those angles are congruent."
Therefore, Triangle ABD is isosceles triangle.
Similarly consider triangle ACD,
By angle sum property,
Therefore,
Therefore, AC = CD
"By the converse of the base angles theorem, which states that if two angles of a triangle are congruent, then sides opposite those angles are congruent."
Therefore, Triangle ADC is isosceles triangle.