In parallelogram ABCD, the angle bisector of ∠A bisects BC. Will angle bisector of B also bisect AD? Give reason.
Solution:
Given, ABCD is a parallelogram.
The angle bisector of ∠A bisects BC.
We have to determine if the angle bisector of B also bisects AD.
The angle bisector of A meets BC at its mid-point P.
Now, draw a line that is parallel to the lines AB and CD and passes through P and meets the line AD at Q.
So, Q is the midpoint of AD.
We know that the alternate interior angles are equal.
∠BAP = ∠APQ ------------------------ (1)
Since AP is the bisector of angle A, we get
∠BAP = ∠PAQ ------------------------- (2)
From (1) and (2),
∠APQ = ∠PAQ
We observe that the triangle APQ is an isosceles triangle
So, AQ = PQ -------------------------- (3)
Also, AQ = 1/2 AD -------------------- (4)
PQ = AB ------------------------------- (5)
Substituting (4) and (5) in (3) we get
AB = 1/2 AD
AD = 2AB
This implies that the angle bisector of B also bisects AD.
Therefore, the angle bisector of B also bisects AD.
✦ Try This: Prove that the perpendicular drawn from the vertex of a regular pentagon to the opposite side bisects that side.
☛ Also Check: NCERT Solutions for Class 8 Maths
NCERT Exemplar Class 8 Maths Chapter 5 Problem 181
In parallelogram ABCD, the angle bisector of ∠A bisects BC. Will angle bisector of B also bisect AD? Give reason.
Summary:
In parallelogram ABCD, the angle bisector of ∠A bisects BC. The angle bisector of B will also bisect AD.
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