Theorem of Cyclic Quadrilateral

Q.

In the given figure, ABCD is a cyclic Quadrilateral. O is the centre of the circle and $\angle BOD={160}^{\circ }$, find the measure of $\angle BPD.$

1. ${70}^{\circ }$
2. ${50}^{\circ }$
3. ${100}^{\circ }$
4. ${20}^{\circ }$

Q.

In the adjoining figure AB is a diameter of the circle and $\angle$BCD = ${130}^{\circ }$. What is the value of $\angle$ABD?

1. 30°

2. none of these

3. 50°

4. 40°

Q.

If a flagstaff subtends the same angle at the points ‘A, B, C & D’ on the horizontal plane through its foot, then ‘A, B, C & D’ from a

1. Square

2. Cyclic quadrilateral

3. Rectangle

4. None of these

Q. Draw a pair of tangents to a circle of radius $5cm$ which are inclined to each other at an angle of ${60}^o$.

Q.

In the given figure $\angle$ AOC $={130}^{\circ }$, The value of $\angle$ABC (in degrees) is _______.

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

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

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

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

Q. ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If $\angle DBC={70}^{\circ },\angle BAC={30}^{\circ }$ and AB=BC, then find $\angle ECD.$

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

Q.

ABCD is a parallelogram. If the radius of the circle passing through all the vertices of a parallelogram is 4cm, then the length of the diagonal AC is ___.

Q. ABCD is a parallelogram. If the radius of the circle passing through all the vertices of the parallelogram is 4 cm, then length of AC is .

1. 4 cm
2. 8 cm
3. 12 cm
4. 16 cm

Q. ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If $\angle DBC={70}^{\circ },\angle BAC\text{is}{30}^{\circ }$, find $\angle BCD$ . Further, if AB = BC, find $\angle ECD$. [2 Marks]

Q.

ABCD is a cyclic quadilateral whose diagonals intersect at a point E. If $\angle DBC={70}^{\circ }$ and $\angle BAC={30}^{\circ }$, find $\angle BCD$. Further if $AB=BC,$ find $\angle ECD$.

1. 60^o

2. 50^o

3. 80^o

4. 70^o

Q.

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

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

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

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

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

Statement 5: ABCD is called cyclic quadrilateral.

1. Statement 1 and statement 5 are true

2. Only statement 4 is true

3. Only statement 1 is true

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

Q.

A, B, C, D are 4 non collinear points in a plane such that $\angle$ ACB $=\angle$ ADB, then how many circle(s) can be drawn passing through all 4 points.

1. 3

2. 0

3. 1

4. 2

Q. In the figure, an isosceles triangle $ABC,$ with $AB=AC,$ circumscribes a circle. Prove that the point of contact $P$ bisects the base $BC.$

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.

ABCD is a cyclic quadilateral whose diagonals intersect at a point E. If $\angle DBC={70}^{\circ }$ and $\angle BAC={30}^{\circ }$, find $\angle BCD$. Further if $AB=BC,$ find $\angle ECD$.

1. ${60}^{\circ },{20}^{\circ }$

2. ${70}^{\circ },{40}^{\circ }$

3. ${80}^{\circ },{50}^{\circ }$

4. ${80}^{\circ },{30}^{\circ }$

Q.

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

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

Q.

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

1. $(\frac{120}{2}{)}^{\circ }$

2. $(\frac{120}{3}{)}^{\circ }$

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

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

Q.

ABCD is a cyclic quadilateral whose diagonals intersect at a point E. If $\angle DBC={70}^{\circ }$ and $\angle BAC={30}^{\circ }$, find $\angle BCD$. Further if $AB=BC,$ find $\angle ECD$.

1. ${80}^{\circ },{50}^{\circ }$

2. ${70}^{\circ },{40}^{\circ }$

3. ${60}^{\circ },{20}^{\circ }$

4. ${80}^{\circ },{30}^{\circ }$

Q.

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

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

2. ${80}^{\circ }$

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

4. ${50}^{\circ }$

Q.

In a cyclic quadrilateral if one angle is 60 degrees then the angle opposite to that is ___

1. 120 degrees

2. 60 degrees

3. 180 degrees

4. 30 degrees

Q.

What is the sum of either pair of opposite angles of a cyclic quadrilateral?

1. ${90}^{\circ }$

2. ${360}^{\circ }$

3. ${45}^{\circ }$

4. ${180}^{\circ }$

Q.

The sum of opposite angles in a cyclic quadrilateral is ____ degrees.

1. 90

2. 180

3. 360

4. 120

Q.

In the given figure, ABCD is a cyclic quadrilateral, OB is the radius, PB is the tangent at point B and $\angle OBC={30}^{\circ }$. AOC is a straight line. Then, the value of x is

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

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

3. ${30}^{\circ }$

4. ${90}^{\circ }$