ABCD is a parallelogram. A circle through A, B is so drawn that it intersects AD at P and BC at Q. Prove that P, Q, C and D are concyclic.

Solution:

Given, ABCD is a parallelogram

A circle through A, B is drawn so that it intersects AD at P and BC at Q.

We have to prove that P, Q, C and D are concyclic.

Join PQ

By exterior angle property of a cyclic quadrilateral,

∠1 = ∠A

We know that the opposite angles of a parallelogram are equal

∠A = ∠C

So, ∠1 = ∠C --------------------------- (1)

Now, AD || BC and PQ is a transversal,

We know that the sum of co interior angles on the same side is 180 degrees.

So, ∠C+ ∠D = 180°

From (1),

∠1+ ∠D = 180°

We know that the sum of opposite angles of a cyclic quadrilateral is equal to 180 degrees.

So, the quadrilateral QCDP is cyclic.

Therefore, the points P, Q, C and D are con-cyclic.

✦ Try This: In the given figure ABCD is a cyclic quadrilateral and AD=BC. If ∠A=110, then find the value of ∠B

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 10

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NCERT Exemplar Class 9 Maths Exercise 10.4 Problem 4

ABCD is a parallelogram. A circle through A, B is so drawn that it intersects AD at P and BC at Q. Prove that P, Q, C and D are concyclic.

Summary:

ABCD is a parallelogram. A circle through A, B is so drawn that it intersects AD at P and BC at Q. It is proven that P, Q, C and D are concyclic

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