Sum of Opposite Angles of a Cyclic Quadrilateral
The quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic. Prove it.
ABCD is a cyclic quadrilateral. If ∠BCD = 100∘ and ∠ ABD = 70∘. Find ∠ ADB.
ABCD is cyclic quadrilateral such that AB is a diameter of the circle circumscribing it and ∠ADC=140∘, then ∠BAC is equal to
Prove that :
(i) the parallelogram, inscribed in a circle, is a rectangle.
(ii) the rhombus, inscribed in a circle, is a square.
In a cyclic quadrilateral ABCD, ∠A=(2x+4)∘, ∠B=(y+3)∘, ∠C=(2y+10)∘, ∠D=(4x−5)∘. Find the four angles.
If a side of a cyclic quadrilateral is produced, then prove that the exterior angle is equal to the interior opposite angle. [3 MARKS]
ABCD is a cyclic quadrilateral. Find the angles of the cyclic quadrilateral.
In the given figure, PQRS is a cyclic quadrilateral in a circle with centre O. If ∠PSR=130∘. Find ∠QPR.
In the given figure, ∠ BAD = 65o, ∠ ABD = 70o and ∠ BDC = 45o, Find :
(i) ∠ BCD
(ii) ∠ ACB
Hence, show that AC is a diameter.
In parallelogram LOST, SN⊥OL and SM⊥LT. Find ∠STM, ∠SON and ∠NSM.
Calculate the measure of ∠AOC.
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∘.
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.
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.
Which of the following figures satisfy the following property?
- Only one pair of sides are parallel.
In the given figure, RS is a diameter of the circle. NM is parallel to RS and ∠ MRS = 29o. Calculate :
(i) ∠ RNM,
(ii) ∠ NRM.
Both the pairs of opposite angles of a quadrilateral are equal and supplementary. Find the measure of each angle.
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.
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.
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.
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.
In cyclic quadrilateral ABCD
∠BAC=30∘ and ∠CBD=70∘; find :
(i) ∠BCD (ii) ∠BCA
(iii) ∠ABC (iv) ∠ADC
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.
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]
In a cyclic quadrilateral ABCD, ∠ A : ∠ C = 3 : 1 and ∠ B : ∠ D = 1 : 5; find each angle of the quadrilateral.
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.