An angle bisector of a triangle is also the altitude. Prove that the triangle is isosceles.

Question

An angle bisector of a triangle is also the altitude. Prove that the triangle is isosceles in statement reason format with all of the statements and reasons.

Answer 1

Let ABC be a triangle. AD is the altitude from A to BC.

The altitude also is an angle bisector (given).

Consider the two triangles ABD and ADC.

1. Angle BAD = Angle CAD.
2. Side AD is common (altitude)
3. Angle ADB = ADC = 90°. So far 2 sets of angles are equal.

By ASA rule, the 2 triangles are congruent. AB=AC and BD=DC. The triangles are isosceles.

Learn:

1. Congruent triangle rules - SSS, SAS, ASA, RHS.
2. Angle bisector theorem, perpendicular bisector theorem
3. Isosceles triangle theorem
4. Similar triangles rules etc. to solve triangle problems in Geometry