In the given figure, $A B$ is a diameter of the circle. PQ is a chord such that $\angle \mathrm{BAP}=\angle \mathrm{ABQ}$. Prove that ABQP is a cyclic trapezium.
View full solution →Question types
Angle and Cyclic Properties of a Circles question types
40 questions across 4 question groups — pick any mix to generate a Mathematics paper with step-by-step answer keys.
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Sample Questions
Angle and Cyclic Properties of a Circles questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
In the figure, A, D, B, C are four points on the circumference of a circle with centre O . arc $\mathrm{AB}=2$ arc BC and $\angle \mathrm{AOB}=108^{\circ}$. Calculate in degrees
(a) $\angle \mathrm{ACB}$
(b) $\angle \mathrm{CAB}$
(c) $\angle \mathrm{ADB}$.
View full solution →(a) $\angle \mathrm{ACB}$
(b) $\angle \mathrm{CAB}$
(c) $\angle \mathrm{ADB}$.
In the given figure, $\angle \mathrm{BAD}=65^{\circ}, \angle \mathrm{ABD}=70^{\circ}$ and $\angle \mathrm{BDC}=45^{\circ}$. Find
(a) $\angle \mathrm{BCD}$
(b) $\angle \mathrm{ADB}$. Hence, show that AC is a diameter of the circle.
View full solution →(a) $\angle \mathrm{BCD}$
(b) $\angle \mathrm{ADB}$. Hence, show that AC is a diameter of the circle.
In the given figure, ABCD is a cyclic quadrilateral. O is the centre of the circle. If $\angle C O D=120^{\circ}$ and $\angle B A C=30^{\circ}$, find $\angle B O C$ and $\angle B C D$.
View full solution →In the given figure, AB is a diameter of the circle. $\angle \mathrm{BDC}=20^{\circ}$ and $\angle C B D=25^{\circ}$. Find $\angle A E D$ and $\angle A C D$.
View full solution →In the figure, O is the centre of the circle. Prove that x + y = z.
Two circles intersect at P and Q. Through P, a straight line APB is drawn to meet the circles in A and B. Through Q, a straight line is drawn to meet the circles at C and D. Prove that AC is parallel to BD.
In the figure, $O$ is the centre of the circle.
$\angle \mathrm{AOE}=150^{\circ}, \angle \mathrm{DAO}=51^{\circ} .$
Calculate the measures of $\angle \mathrm{BEC}$ and $\angle \mathrm{EBC}$.
View full solution →$\angle \mathrm{AOE}=150^{\circ}, \angle \mathrm{DAO}=51^{\circ} .$
Calculate the measures of $\angle \mathrm{BEC}$ and $\angle \mathrm{EBC}$.
In the given figure, AC is the diameter of the circle with centre $\mathrm{O} . \mathrm{CD}$ and BE are parallel, $\angle \mathrm{AOB}=80^{\circ}$ and $\angle \mathrm{ACE}=10^{\circ}$.
Calculate (a) $\angle \mathrm{BEC}$ (b) $\angle \mathrm{BCD}$ (c) $\angle \mathrm{CED}$.
View full solution →Calculate (a) $\angle \mathrm{BEC}$ (b) $\angle \mathrm{BCD}$ (c) $\angle \mathrm{CED}$.
Prove that any four vertices of a regular pentagon are concyclic.
In the given figure, the value of $x$ is :
- A$86^{\circ}$
- B$84^{\circ}$
- C$82^{\circ}$
- D$80^{\circ}$
If $O$ is the circumcentre of $\triangle A B C$ and $O D \perp B C$, then $\angle \mathrm{BOD}=$
- A$\angle \mathrm{A}$
- B$\angle \mathrm{B}$
- C$\angle \mathrm{C}$
- Dnone of these
In the figure, chord ED is parallel to the diameter AC of the circle. Given, $\angle \mathrm{CBE}=65^{\circ}$, then $\angle \mathrm{DEC}=$
- A$35^{\circ}$
- B$30^{\circ}$
- C$25^{\circ}$
- D$20^{\circ}$
In the figure, $O$ is the centre of the circle. The angle subtended by the arc BCD at the centre is $140^{\circ}, \mathrm{BC}$ is produced to P , then $\angle \mathrm{BAD}+\angle \mathrm{BCD}=$
- A$160^{\circ}$
- B$170^{\circ}$
- C$180^{\circ}$
- D$210^{\circ}$
In the figure, if $\angle \mathrm{AOB}=90^{\circ}$ and $\angle \mathrm{ABC}=30^{\circ}$, then $\angle \mathrm{CAO}$ is equal to :
- A$30^{\circ}$
- B$45^{\circ}$
- C$90^{\circ}$
- D$60^{\circ}$
Assertion (A) : In the figure, if AB is a diameter of the circle, then $\angle \mathrm{AED}+\angle \mathrm{DCB}=270^{\circ}$.
Reason (R) : Angle in a semi-circle is a right angle.
Reason (R) : Angle in a semi-circle is a right angle.
- AA is true, R is false.
- BA is false, R is true.
- CBoth A and R are true, and R is the correct reason for A .
- ✓Both A and R are true, and R is incorrect reason for A .
Answer: D.
View full solution →Assertion (A) : In the figure, if O is the centre of the circle, then $\angle \mathrm{CBD}=60^{\circ}$.
Reason (R) : Exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
Reason (R) : Exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
- AA is true, R is false.
- ✓A is false, R is true.
- CBoth A and R are true, and R is the correct reason for A .
- DBoth A and R are true, and R is incorrect reason for A .
Answer: B.
View full solution →Assertion (A) : In the figure, $\angle \mathrm{ACD}=35^{\circ}$.
Reason (R) : Opposite angles of a cyclic quadrilateral are equal.
Reason (R) : Opposite angles of a cyclic quadrilateral are equal.
- ✓A is true, R is false.
- BA is false, R is true.
- CBoth A and R are true, and R is the correct reason for A .
- DBoth A and R are true, and R is incorrect reason for A .
Answer: A.
View full solution →Assertion (A) : In the figure, $O$ is the centre of the circle. The value of $\angle \mathrm{ACB}$ is also $40^{\circ}$.
Reason (R): The angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of the circle.
Reason (R): The angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of the circle.
- AA is true, R is false.
- ✓A is false, R is true.
- CBoth A and R are true, and R is the correct reason for A .
- DBoth A and R are true, and R is incorrect reason for A .
Answer: B.
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