Prove that a cyclic parallelogram is a rectangle

Solution:

The sum of either pair of opposite angles of a cyclic quadrilateral is 180°. Using this fact, we can show each angle of a cyclic parallelogram as 90°, proving the statement it is a rectangle.

Let ABCD be the cyclic parallelogram.

We know that opposite angles of a parallelogram are equal.

∠A = ∠C and ∠B = ∠D   ... (1)

We know that the sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

∠A + ∠C = 180°

∠A + ∠A = 180° (From equation (1))

2∠A = 180°

∠A = 90°

We know that if one of the interior angles of a parallelogram is 90°, all the other angles will also be equal to 90°.

Since all the angles in the parallelogram are 90°, we can say that parallelogram ABCD is a rectangle.

☛ Check: NCERT Solutions Class 9 Maths Chapter 10

Video Solution:

Prove that a cyclic parallelogram is a rectangle

Maths NCERT Solutions Class 9 Chapter 10 Exercise 10.5 Question 12

Summary:

We know that if one of the interior angles of a parallelogram is 90°, all the other angles will also be equal to 90°. Since all the angles in the parallelogram are 90°, we can say that parallelogram ABCD is a rectangle. Hence a cyclic parallelogram is a rectangle.

☛ Related Questions:

• If the non-parallel sides of a trapezium are equal, prove that it is cyclic.
• Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D, P and Q respectively (see Fig. 10.40). Prove that ∠ACP = ∠QCD.
• If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.
• ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠CAD = ∠CBD