Sum of Opposite Angles of a Cyclic Quadrilateral

Q.

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

Q.

ABCD is a cyclic quadrilateral. If $\angle$BCD = 100${}^{\circ }$ and $\angle$ ABD = 70${}^{\circ }$. Find $\angle$ ADB.

1. 30°

2. 40°

3. 60°

4. 50°

Q.

$ABCD$ is cyclic quadrilateral such that $AB$ is a diameter of the circle circumscribing it and $\angle ADC={140}^{\circ },$ then $\angle BAC$ is equal to

(A) ${80}^{\circ }$
(B) ${50}^{\circ }$
(C) ${40}^{\circ }$
(D) ${30}^{\circ }$

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, $\angle A=(2x+4{)}^{\circ },\angle B=(y+3{)}^{\circ },\angle C=(2y+10{)}^{\circ },\angle D=(4x-5{)}^{\circ }$. 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 $\angle PSR={130}^{\circ }$. Find $\angle QPR$.​​​​​​

1. ${50}^{\circ }$

2. ${40}^{\circ }$

3. ${60}^{\circ }$

4. ${30}^{\circ }$

Q.

In the given figure, $\angle$ BAD = ${65}^o$, $\angle$ ABD = ${70}^o$ and $\angle$ BDC = ${45}^o$, Find :

(i) $\angle$ BCD

(ii) $\angle$ ACB

Hence, show that AC is a diameter.

Q. Question 153
In parallelogram LOST, $SN\perp OL$ and $SM\perp LT$. Find $\angle STM,\angle SON$ and $\angle NSM$.

Q.

Calculate the measure of $\angle AOC.$

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 : $\angle APB={60}^{\circ }.$

Q.

ABCD is a cylic quadrilateral in which BC is parallel to AD, angle ADC = ${110}^o$ and angle BAC = ${50}^o$. 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 $\angle$ APB = ${75}^o$ and $\angle$BCD = ${40}^o$, find :

(i) $\angle$ AOB,

(ii) $\angle$ ACB,

(iii) $\angle$ ABD,

(iv) $\angle$ 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 $\angle$ MRS = ${29}^o$. Calculate :

(i) $\angle$ RNM,

(ii) $\angle$ 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 $\angle$ DEF = ${110}^o$, calculate :

(i) $\angle$ AEF, (ii) $\angle$ FAB.

Q.

In a cyclic-quadrilateral PQRS, angle $PQR={135}^{\circ }.$ Sides SP and RQ produced meet at point A : whereas sides PQ and SR produced meet at point B.

If $\angle A:\angle B=2:1;$ 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 $\angle DCF:\angle F:\angle E=3:5:4,$ 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 $\angle$ APB = ${60}^o$.

Q.

In cyclic quadrilateral ABCD

AD =BC,

$\angle BAC={30}^{\circ }and\angle CBD={70}^{\circ };$ find :

(i) $\angle BCD$ (ii) $\angle BCA$

(iii) $\angle ABC$ (iv) $\angle ADC$

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

1. True
2. 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) $\angle$ ADC (ii) $\angle$ BDA

(iii) $\angle$ ABC, (iv) $\angle$ 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, $\Delta ABD$ is equilateral]

Q.

In a cyclic quadrilateral ABCD, $\angle$ A : $\angle$ C = 3 : 1 and $\angle$ B : $\angle$ 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.