ABCD is a parallelogram. The bisector of angle A intersects CD at X and bisector of angle C intersects AB at Y. Is AXCY a parallelogram? Give reason.
Solution:
Given, ABCD is a parallelogram.
The bisector of angle A intersects CD at X and bisector of angle C intersects AB at Y.
We have to determine if AXCY is a parallelogram.
We know that the opposite angles are equal in a parallelogram.
So, ∠A = ∠C
Dividing by 2 on both sides,
1/2 ∠A = 1/2 ∠C
from the figure,
∠1 = ∠2 ------------------ (1)
Since AB || CD with CY as transversal, the alternate interior angles are equal.
So, ∠2 = ∠3
From (1),
∠1 = ∠3
Since the corresponding angles are equal, AX || YC ------ (a)
Similarly, in parallelogram ABCD, AB || DC
So, AY || XC ------------------------------------ (b)
From (a) and (b),
The opposite sides are parallel.
Therefore, AXCY is a parallelogram.
✦ Try This: In a quadrilateral ABCD, ∠A = ∠B = ∠C and ∠D is equal to 150°. Find the values of A, B and C.
☛ Also Check: NCERT Solutions for Class 8 Maths
NCERT Exemplar Class 8 Maths Chapter 5 Problem 176
ABCD is a parallelogram. The bisector of angle A intersects CD at X and bisector of angle C intersects AB at Y. Is AXCY a parallelogram? Give reason.
Summary:
ABCD is a parallelogram. The bisector of angle A intersects CD at X and bisector of angle C intersects AB at Y. AXCY is a parallelogram.
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