# Sum of Opposite Angles of a Cyclic Quadrilateral

## Trending Questions

**Q.**

The quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic. Prove it.

**Q.**

ABCD is a cyclic quadrilateral. If ∠BCD = 100∘ and ∠ ABD = 70∘. Find ∠ ADB.

30°

40°

60°

50°

**Q.**

ABCD is cyclic quadrilateral such that AB is a diameter of the circle circumscribing it and ∠ADC=140∘, then ∠BAC is equal to

(A) 80∘

(B) 50∘

(C) 40∘

(D) 30∘

**Q.**

Prove that :

(i) the parallelogram, inscribed in a circle, is a rectangle.

(ii) the rhombus, inscribed in a circle, is a square.

**Q.**The angles of a cyclic quadrilateral ABCD are <A = 6x + 10 , <B = 5x , <C = x + y and <D = 3y - 10 . find all the angles of the cyclic quadrilateral

**Q.**

In a cyclic quadrilateral ABCD, ∠A=(2x+4)∘, ∠B=(y+3)∘, ∠C=(2y+10)∘, ∠D=(4x−5)∘. Find the four angles.

**Q.**

If a side of a cyclic quadrilateral is produced, then prove that the exterior angle is equal to the interior opposite angle. [3 MARKS]

**Q.**

Question 8

ABCD is a cyclic quadrilateral. Find the angles of the cyclic quadrilateral.

**Q.**

In the given figure, PQRS is a cyclic quadrilateral in a circle with centre O. If ∠PSR=130∘. Find ∠QPR.

50∘

40∘

60∘

30∘

**Q.**

In the given figure, ∠ BAD = 65o, ∠ ABD = 70o and ∠ BDC = 45o, Find :

(i) ∠ BCD

(ii) ∠ ACB

Hence, show that AC is a diameter.

**Q.**Question 153

In parallelogram LOST, SN⊥OL and SM⊥LT. Find ∠STM, ∠SON and ∠NSM.

**Q.**

Calculate the measure of ∠AOC.

**Q.**Prove that, any rectangle is a cyclic quadrilateral.

**Q.**

In the following figure, AB is the diameter of a circle with centre O and CD is the chord with length euqal to radius OA.

If AC produced and BD produced meet at point P; show that : ∠APB=60∘.

**Q.**

ABCD is a cylic quadrilateral in which BC is parallel to AD, angle ADC = 110o and angle BAC = 50o. Find angle DAC and angle DCA.

**Q.**

In the given figure, the centre O of the small circle lies on the circumference of the bigger circle. If ∠ APB = 75o and ∠BCD = 40o, find :

(i) ∠ AOB,

(ii) ∠ ACB,

(iii) ∠ ABD,

(iv) ∠ ADB.

**Q.**Question 8

Which of the following figures satisfy the following property?

- Only one pair of sides are parallel.

**Q.**

In the given figure, RS is a diameter of the circle. NM is parallel to RS and ∠ MRS = 29o. Calculate :

(i) ∠ RNM,

(ii) ∠ NRM.

**Q.**Question 166

Both the pairs of opposite angles of a quadrilateral are equal and supplementary. Find the measure of each angle.

**Q.**

In the following figure, AD is the diameter of the circle with centre O. Chords AB, BC and CD are equal. If ∠ DEF = 110o, calculate :

(i) ∠ AEF, (ii) ∠ FAB.

**Q.**

In a cyclic-quadrilateral PQRS, angle PQR=135∘. Sides SP and RQ produced meet at point A : whereas sides PQ and SR produced meet at point B.

If ∠A : ∠B=2 :1; find angles A and B.

**Q.**12. Prove that the sum of opposite sides of quadrilateral circumscribed about a circle is always equal. i.e. if a quadrilateral ABCD is circumscribed about a circle the

**Q.**

ABCD is a cyclic quadrilateral. Sides AB and DC produced meet at point E; whereas sides BC and AD produced meet at point F.

If ∠DCF : ∠F : ∠E=3 : 5 : 4 , find the angles of the cyclic quadrilateral ABCD.

**Q.**

In a circle, with centre O, a cyclic quadrilateral ABCD is drawn with AB as a diameter of the circle and CD equal to radius of the circle. If AD and BC produced meet at point P; show that ∠ APB = 60o.

**Q.**

In cyclic quadrilateral ABCD

AD =BC,

∠BAC=30∘ and ∠CBD=70∘; find :

(i) ∠BCD (ii) ∠BCA

(iii) ∠ABC (iv) ∠ADC

**Q.**In cyclic quadrilateral, the sum of either pair of opposite angles is equal to 90 degrees

- True
- False

**Q.**

In the given figure, BD is a side of a regular hexagon, DC is a side of a regular pentagon and AD is a diameter. Calculate :

(i) ∠ ADC (ii) ∠ BDA

(iii) ∠ ABC, (iv) ∠ AEC.

**Q.**Question 179

ABCD is a rhombus such that the perpendicular bisector of AB passes through D. Find the angles of the rhombus.

[Hint Join BD. Then, ΔABD is equilateral]

**Q.**

In a cyclic quadrilateral ABCD, ∠ A : ∠ C = 3 : 1 and ∠ B : ∠ D = 1 : 5; find each angle of the quadrilateral.

**Q.**

ABCD is a cyclic quadrilateral in which AB and DC on being produced, meet at P such that PA = PD. Prove that AD is parallel to BC.