Q) Prove that a cyclic parallelogram is a rectangle. 

9th Class Maths – NCERT Important Questions

Ans:

Step 1: Let’s make a diagram for better understanding of the question:

Here ABCD is the cyclic parallelogram and O is the center of the circle circumscribing it.

Step 2:

We are given that ABCD is a parallelogram and

∵ Opposite angles of a parallelogram are always equal.

∴ ∠ DAB = ∠ DCB ………. (i)

and ∠ ABC = ∠ ADC ……….. (ii)

Step 3: Next, we are given that ABCD is cyclic quadrilateral

∵ In a cyclic quadrilateral, Opposite angles are supplementary

∴ ∠ DAB + ∠ DCB  = 180⁰ ………… (iii)

and ∠ ABC + ∠ ADC = 180⁰ ………….. (iv)

Step 4: By substituting equation (i) in equation (iii), we get:

∵ ∠ DAB + ∠ DCB  = 180⁰

∴ ∠ DAB + ∠ DAB  = 180⁰

∴ 2 ∠ DAB = 180⁰

∴ ∠ DAB = 90⁰

Similarly, By substituting equation (ii) in equation (iv), we get ∠ ABC = 90⁰

Step 5: So now 2 angles of given cyclic parallelogram ABCD are 90⁰

Now, when one or two angles of a parallelogram is 90⁰, then all the other angles are also 90⁰.

And when all 4 angles of parallelogram ABCD are 90⁰,

Then ABCD is a rectangle.

Hence Proved!

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