Sum of Opposite Angles of a Cyclic Quadrilateral

Q.

Prove that a cyclic parallelogram is a rectangle.

Q. Prove that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Q.

In the figure, ABCD is a cyclic quadrilateral in which AC and BD are its diagonals. If $\angle DBC={55}^{\circ }and\angle BAC={45}^{\circ },$ find $\angle BCD$.

Q.

In the given figure, O is the centre of a circle and $\angle AOB={140}^{\circ }.Then\angle ACB=?$

(a) ${70}^{\circ }$

(b) ${80}^{\circ }$

(c) ${110}^{\circ }$

(d) ${40}^{\circ }$

Q. Prove that the quadrilateral formed (if possible) by the internal angle bisectors of any quadrilateral is cyclic.

Q.

In a cyclic quadrilateral ABCD, if $AB||CD$ and $\angle B={70}^{\circ },$ find the remaining angles.

Q.

In the given figure, O is the centre of a circle and $\angle OAB={50}^{\circ },Then,\angle BOD=?$

(a) ${130}^{\circ }$

(b) ${50}^{\circ }$

(c) ${100}^{\circ }$

(d) ${80}^{\circ }$

Q.

In the given figure, $\angle ABD={54}^{\circ }\text{and}\angle BCD={43}^{\circ },$, calculate

(i) $\angle ACD$

(ii) $\angle BAD$

(iii) $\angle BDA$.

Q.

In the given figure, ABCD and ABEF are two cyclic quadrilateral. If $\angle BCD={110}^{\circ }then\angle BEF=?$

(a) ${55}^{\circ }$

(b) ${70}^{\circ }$

(c) ${90}^{\circ }$

(d) ${110}^{\circ }$

Q.

In the figure, two circles intersect at A and B. The centre of the smaller circle is O and it lies on the circumference of the larger circle. If $\angle APB={70}^o$, find $\angle ACB$.

Q.

In the figure, ABCD is a cyclic quadrilateral, Find the value of x.

Q.

PQRS is a cyclic quadrilateral such that PR is a diameter of the circle. If $\angle QPR={67}^{o}and\angle SPR={72}^o$, then $\angle QRS=$

1. ${41}^o$

2. ${23}^o$

3. ${67}^o$

4. ${18}^o$

Q.

In the given figure, ABCD is a cyclic quadrilateral in which DC is produced to E and CF is drawn parallel to AB such that $\angle ADC={95}^{\circ }and\angle ECF={20}^{\circ }Then,\angle BAD=?$.

(a) ${95}^{\circ }$

(b) ${85}^{\circ }$

(c) ${105}^{\circ }$

(d) ${75}^{\circ }$

Q.

In the figure, ABCD is a cyclic quadrilateral in which $\angle BAD={75}^o$, $\angle ABD={58}^o$ and $\angle ADC={77}^o$, AC and BD intersect at P. Then, find $\angle DPC$.

Q.

In the given figure, ABCD is a cyclic quadrilateral in which AB||DC and $\angle BAD={100}^{\circ }.Then,\angle ABC=?$

(a) ${80}^{\circ }$

(b) ${100}^{\circ }$

(c) ${50}^{\circ }$

(d) ${40}^{\circ }$

Q.

In the given figure, POQ is a diameter and PQRS is a cyclic quadrilateral. If $\angle PSR={150}^{\circ },\text{find}\angle RPQ$.

Q.

ABCD is a cyclic trapezium with AD || BC. If $\angle B={70}^{\circ }$, determine other three angles of the trapezium.

Q.

If $ABCD$ is a cyclic quadrilateral, then the value of $\cos A-\cos B+\cos C-\cos D$ is

1. $0$

2. $1$

3. $2(\cos B–\cos D)$

4. $2(\cos A–\cos C)$

Q.

In the figure, $\Delta ABC$ is an equilateral triangle. Find m$\angle BEC$.

Q.

If ABCD is a cyclic quadrilateral in which $AD||BC$. Prove that $\angle B=\angle C$.

Q.

In the given figure, ABCD is a cyclic quadrilateral whose diagonals intersect at P such that $\angle DBC={60}^{\circ }and\angle BAC={40}^{\circ }.\text{Find}\left(i\right)\angle BCD,\left(ii\right)\angle CAD$.

Q.

In the figure, ABCD is a quadrilateral inscribed in a circle with centre O. CD produced to E such that $AED={95}^o$ and $\angle OBA={30}^o$. Find $\angle OAC$.

Q.

In the given figure, ABCD is a cyclic quadrliateral in which AE is drawn parallel to CD, and BA is produced to F. If $\angle ABC={92}^{\circ }and\angle FAE={20}^{\circ },\text{find}\angle BCD.$

Q.

In the figure, if $\angle BAC={60}^{o}and\angle BCA={20}^o,find\angle ADC$

Q.

In the figure, O is the centre of the circle. If $\angle$OBC = 25${}^{\circ }$, then $\angle$BAC in degree is equal to:

1. 150

2. 25

3. 30

4. 65

Q.

ABCD is a cyclic quadrilateral in which :

(i) BC || AD, $ADC={110}^{o}and\angle BAC={50}^o$. Find $\angle DAC$.

(ii) $\angle DBC={80}^{o}and\angle BAC={40}^o.$ Find $\angle BCD$.

(iii) $\angle BCD={100}^{o}andABD={70}^o$. Find $\angle ADB$.

Q.

In the given figure, $ABCD$ is a quadrilateral in which $AD=BC$ and $\angle ADC=\angle BCD$. Show that the points $A,B,C,D$ lie on a circle.

Q. 2. How to prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Q. What is the sum of opposite angles of a cyclic quadrilateral?

1. $180°$
2. $270°$
3. $360°$
4. $90°$

Q. ABCD is a cyclic quadrilateral such that ∠ADB = 30° and ∠DCA = 80°, then ∠DAB =

(a) 70°

(b) 100°

(c) 125°

(d) 150°