Question

ABC is an isosceles triangle in which AB=AC. Also, D is a point such that BD=CD. Prove that AD bisects angle A and angle D.

Please give a full explanation

Click on the above attachment to see the diagram.​

Answer 1

Answer:

We know that

AC = AB (Given)

BD = CD (Given)

Hence, using these two properties, we can say that quadrilateral ABCD is a kite.

One of the properties of a kite is that the main diagonal (AD) is the perpendicular bisector of the cross diagonal (BC).

Hence, we can say that

Diagonal AD bisects Diagonal BC.

Hence, OB = OD (O is the intersection point of the 2 diagonals)

Therefore in ΔABO and ΔACO

AB = AC (Given)

∠ABC = ∠ACB (Isosceles triangle property where opposite angles of equal sides are equal)

OB = OB

Therefore, we can say that ΔABO ≅ ΔACO (SAS)

∠BAD = ∠CAD (C.P.C.T)

Hence we can say that AD bisects ∠A.

Similarly in ΔBOD and ΔCOD

BD = CD (Given)

∠DBO = ∠DCO (Isosceles triangle property where opposite angles of equal sides are equal)

OB = OC (Proved)

Hence, we can say that ΔBOD ≅ ΔCOD (SAS)

∠BDO = ∠CDO (C.P.C.T)

Hence, we can say that AD bisects ∠D as well.

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