Theorem 5: Arcs That Subtend Equal Angle at Center Are Equal

Q.

In the given figure, $O$ is the centre of the circle. Find the value of $x$.

1. 30°

2. 40°

3. 50°

4. 100°

Q.

The circumference of a circle with centre O is divided into three arcs APB, BQC and CRA such that

$\frac{arcAPB}{2}=\frac{arcBQC}{3}=\frac{arcCRA}{4}$. $\angle$ Find $\angle$ COA.

1. This situation holds no signifiance

2. OP need not be radius of circle

3. OP is perpendicular to XY

4. The angle between OP and XY is variable

Q.
In the given figure AB is the diameter of the circle. Find the value of y.

1. ${40}^{\circ }$
2. ${90}^{\circ }$
3. ${30}^{\circ }$
4. ${50}^{\circ }$

Q.

What is the relation between $\widehat{AB}$ and $\widehat{CD}$?

1. $\widehat{AB}>\widehat{CD}$

2. $\widehat{AB}<\widehat{CD}$

3. $\widehat{AB}=\widehat{CD}$

4. $\widehat{AB}=\frac{1}{2}\widehat{CD}$

Q. What is the value of $\angle DBC$ in the above figure?

1. ${20}^{\circ }$
2. ${60}^{\circ }$
3. ${70}^{\circ }$
4. ${100}^{\circ }$

Q. From the figure, find all the four angles of the cyclic quadrilateral.

Q. PQRS is a parallelogram and PA, SB, RC, QD, are angle bisectors. If PQ = QD = 6 units, find $\angle PQR.$

1. 30${}^{\circ }$
2. 60${}^{\circ }$
3. 120${}^{\circ }$
4. Indeterminate

Q. In the given figure, If O is the centre of the circle and the minor arc corresponding to chord MN is equal to the minor arc corresponding to chord PQ, then which of the following is true?

1. ML = QR
2. NL < QR
3. ML > PR
4. NL > PR

Q. $\square ABCD$ is a cyclic quadrilateral. $m\left(arcABC\right)={220}^0$.

1. 130
2. 70
3. 110
4. 60

Q.

In the figure, lines AB and CD intersect at O.If $\angle AOC+\angle BOE={70}^{\circ }$ and $\angle BOD={40}^{\circ }$, find $\angle BOE$ and reflex $\angle COE$ . [4 MARKS]

Q. If a diameter of a circle bisects each of the two chords of a circle then prove that the chords are parallel.

Q. If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D. Prove that AB = CD (see figure)

Q. In figure, line $PR$ touches the circle at point $Q$. Answer the following with the help of the figure:
What is the sum of $\angle TAQ$ and $\angle TSQ$?

Q. $ABCD$ is a cyclic quadrilateral with $AD$ parallel to $BC$. $\angle DCB={65}^o$. The magnitude of $\angle CBE$ is:

1. ${100}^o$
2. ${110}^o$
3. ${120}^o$
4. ${115}^o$

Q. The measure of central angle is ${120}^o$. Find the measure of the arc it intercepts.

1. ${60}^o$
2. ${70}^o$
3. ${50}^o$
4. None of the above

Q. In the adjoining figure, $ABCD$ is a cyclic quadrilateral in which $\angle BCD={100}^o$ and $\angle ABD={50}^o$. Find $\angle ADB$.

Q.

In the given figure, if $\angle AOC={130}^{\circ },$ then the value of $\angle ABC$ = _______.

1. ${95}^{\circ }$

2. ${115}^{\circ }$

3. ${50}^{\circ }$

4. ${100}^{\circ }$

Q. ABCD is a cyclic quadrilateral whose side AB is a diameter of the circle through A, B, C and D. If $\angle ADC={130}^{\circ }$, then find $\angle BAC$.

1. ${60}^{\circ }$
2. ${40}^{\circ }$
3. ${50}^{\circ }$
4. ${25}^{\circ }$

Q.

What is the relation between $\widehat{BC}$ and $\widehat{AD}$?

1. $\widehat{BC}>\widehat{AD}$

2. $\widehat{BC}=\widehat{AD}$

3. $\widehat{BC}<\widehat{AD}$

4. $\widehat{BC}=2\widehat{AD}$

Q.

The circumference of a circle with centre O is divided into three arcs APB, BQC and CRA such that

$\frac{arcAPB}{2}=\frac{arcBQC}{3}=\frac{arcCRA}{4}$. Find $\angle BOA.$

1. ${45}^{\circ }$

2. ${60}^{\circ }$

3. ${80}^{\circ }$

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

Q.

If ABCD is a cyclic quadrilateral with AB as the diameter of the circle and $\angle ACD={50}^{\circ },$ then $\angle BAD$ = ___.

1. ${80}^{\circ }$

2. ${70}^{\circ }$

3. ${50}^{\circ }$

4. ${40}^{\circ }$

Q. Find the value of $x$ in the given figure if $\angle AOB={70}^{\circ }$ and $AB=CD.$

1. ${80}^{\circ }$
2. ${40}^{\circ }$
3. ${60}^{\circ }$
4. ${70}^{\circ }$

Q. PQRS is a cyclic quadrilateral whose diagonals intersect at A. If $\angle SQR={80}^{\circ }$ and $\angle QPR={30}^{\circ },$ then find $\angle SRQ.$

1. ${70}^{\circ }$
2. ${60}^{\circ }$
3. ${80}^{\circ }$
4. ${50}^{\circ }$

Q. In a quadrilateral $PQRS,\angle P,\angle Q,\angle R$ and $\angle S$ are interior angles. If $\angle P:\angle Q:\angle R:\angle S=1:2:3:4$, then which angle is equal to ${144}^{\circ }$.

1. $\angle P$
2. $\angle Q$
3. $\angle R$
4. $\angle S$

Q. In the fig. Straight lines $AB$ & $CD$ pass through centre $O$. If angle $\angle DCE={40}^o$ & $\angle AOD={75}^o$.
Find $\angle CDE$ & $\angle OBE$.