Properties of Cyclic Quadrilateral

Q.

PQRS is a cyclic quadrilateral $\angle P=3x,\angle Q=y,\angle R=x,\angle S=5y$. Find x and y.

1. $x={30}^{\circ },y={45}^{\circ }$

2. $x={30}^{\circ },y={30}^{\circ }$

3. $x={45}^{\circ },y={30}^{\circ }$

4. $x={45}^{\circ },y={45}^{\circ }$

Q.

The sum of a pair of opposite angles of a cyclic quadrilateral is 180 degrees .

1. True

2. False

Q.

Prove that if a pair of opposite angles of a quadrilateral is supplementary, then the quadrilateral is cyclic. [4 MARKS]

Q. In the given figure, AC is a diameter of the circle. Find the value of $x.$

1. ${100}^{\circ }$
2. ${126}^{\circ }$
3. ${136}^{\circ }$
4. ${154}^{\circ }$

Q.

In the given figure, O is the centre of the circle and $\angle ABC={55}^{\circ }$. Calculate the values of x and y.

1. $x={110}^{\circ }$

   $y={225}^{\circ }$

2. $x={110}^{\circ }$

   $y={125}^{\circ }$

3. $x={100}^{\circ }$

   $y={125}^{\circ }$

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

   $y={120}^{\circ }$

Q. In the given figure, ∠AOD = 135^∘ then ∠BOC is equal to
(a) 25^∘
(b) 45^∘
(c) 52.5^∘
(d) 62.5^∘

Q.

If quadrilateral ABCD is cyclic, then find the angle subtended at the centre by vertex B and D i.e. $\angle BOD.$

1. ${50}^{\circ }$
2. ${45}^{\circ }$
3. ${30}^{\circ }$
4. ${60}^{\circ }$

Q. Which of the following pairs of opposite angles can a cyclic quadrilateral have?

1. 90° and 45°
2. 60° and 120°
3. 45° and 135°
4. 70° and 120°
5. 25° and 155°
6. 100° and 80°
7. 88° and 91°
8. 130° and 50°

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

1. ${30}^{\circ }$
2. ${40}^{\circ }$
3. ${50}^{\circ }$
4. ${60}^{\circ }$

Q.

AB is the diameter of the circle.CD is parallel to AB, $\angle CAB={32}^{\circ }$. Find $\angle ADC$.

1. ${110}^{\circ }$

2. ${105}^{\circ }$

3. ${122}^{\circ }$

4. ${140}^{\circ }$

Q.

$\text{In the given figure BD = DC and}\angle CBD={30}^{\circ }\text{Find}\left(\angle BAC\right)$

1. ${60}^{\circ }$

2. ${180}^{\circ }$

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

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

Q.

Prove that if a pair of opposite angles of a quadrilateral are right, then a circle can be drawn through all four of its vertices.

Q.

In the figure below XYQZ is a quadrilateral with its vertices on a circle and XYZ is an equilateral triangle. Find the value of $\angle$ YQZ.

1. ${120}^{\circ }$

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

3. ${110}^{\circ }$

4. ${60}^{\circ }$

Q. A quadrilateral is cyclic if it can be inscribed in a circle.

1. True
2. False

Q. The angle of a cyclic quadrilateral ABCD are
<A=(2x+4)°, <B=(y+3)°, <c=(2y+10)°, <D=(4x-5)°.
Find x and y and hence values of four angles.

Q.

In $\Delta$ ABC, AB= AC. P is the mid point of AB and Q is the mid point of AC then:-

1. PQBC is a parallelogram

2. BPCQ is a cyclic quadrilateral

3. PQBC is a square

4. BPCQ is a rectangle

Q.

AB is a diameter of the circle and A, P, B, R are four points on the circle. The lines AP, RB intersect at Q. Find $\angle BPR.$

1. ${28}^{\circ }$
2. ${32}^{\circ }$
3. ${56}^{\circ }$
4. ${62}^{\circ }$

Q. Which of the following pairs of opposite angles can a cyclic quadrilateral have?

1. 90° and 45°
2. 70° and 120°
3. 88° and 91°
4. 25° and 155°
5. 45° and 135°
6. 100° and 80°
7. 72° and 108°
8. 60° and 120°

Q.

In the given circle, with AB as diameter, find the value of x.

[ 1 mark]

Q.

ABCD is a cyclic quadrilateral whose side AB is the diameter of the circle with centre O through A, B, C, D. If $\angle ADC={130}^{\circ }$, Calculate $\angle$BAC.

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

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

3. ${90}^{\circ }$

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

Q.

In the figure below P, Q, R are the points where the incircle of $△ABC$ touches the sides

$\angle AQR$ and $\angle ARQ$ are ______ , ______ respectively.

1. ${40}^{\circ },{70}^{\circ }$

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

3. ${50}^{\circ },{40}^{\circ }$

4. ${70}^{\circ },{140}^{\circ }$

Q.

If ABCD is a cyclic quadrilateral. What will be the value of $x$ (in degrees)?

1. 120°

2. 90°

3. 100°

4. None of these

Q. If all the sides of a quadrilateral are produced in the same order (clockwise or anticlockwise).
Show that quadrilateral formed by the bisectors of exterior angles is cyclic.

Q.

AB is a diameter of the circle and A, P, B, R are four points on the circle. The lines AP, RB intersect at Q. Find $\angle BPR.$

Q.

ABCD is a quadrilateral such that $\angle$ABC+$\angle$ADC =${180}^{\circ }$. Inside the quadrilateral :

Statement 1: the circumcircle of $\Delta$ABC intersects diagonal BD at D.

Statement 2: the circumcircle of $\Delta$ABC intersects BD at $D^{{}^{\prime }}$inside the quadrilateral.

Statement 3: the circumcircle of $\Delta$ABC intersects BD at $D^{{}^{\prime }}$ outside the quadrilateral.

Statement 4: the circumcircle of $\Delta$ABC does not intersect BD at all.

Statement 5: ABCD is called cyclic quadrilateral.

1. Only statement 1 is true

2. Statement 1 and statement 5 are true

3. One of the statement 2 or statement 3 can be true

4. Only statement 4 is true

Q. Choose the correct answer from the alternatives given :
ABCD is a cyclic quadrilateral. Diagonals AC and BD meets at P. If $\angle APB$ = ${110}^{\circ }$ and $\angle CBD$ = ${30}^{\circ }$ , then $\angle ADB$ measures

1. 55${}^{\circ }$
2. 30${}^{\circ }$
3. 80${}^{\circ }$
4. 70${}^{\circ }$

Q.

Which of the following cannot be a cyclic quadrilateral?

1. Parallelogram

2. Rectangle

3. Square

4. Trapezium

Q. In $△$ABC seg AB = seg AC and P is any point on AC. Through C, a line is drawn to intersect BP produced in Q such that $\angle$ABQ = $\angle$ACQ. Prove that $\angle \mathrm{ACQ}=90°+\frac{1}{2}\angle \mathrm{BAC}$

Q. Which of the following pairs of opposite angles can a cyclic quadrilateral have?

1. 45° and 135°
2. 70° and 120°
3. 100° and 80°
4. 25° and 155°
5. 88° and 91°
6. 90° and 45°
7. 72° and 108°
8. 60° and 120°

Q. Given a $△ABC$ with unequal sides. P is the set of all points which is equidistant from B & C and Q is the set of all point which is equidistant from sides AB and AC. Then $n(P\cap Q)$ equals:

1. $1$
2. $2$
3. $3$
4. Infinite