Sum of Pair of Opposite Angles in Quadrilateral

Q. Question 2
In figure, if $\angle AOB={125}^{\circ }$. Then $\angle$ COD is equal to
(A) 62.5°
(B) 45°
(C) 35°
(D) 55°

Q. Question 27
In the trapezium ABCD, the measure of $\angle D$ is
a) ${55}^{\circ }$
b) ${115}^{\circ }$
c) ${135}^{\circ }$
d) ${125}^{\circ }$

Q.

Which of the following cannot be a cyclic quadrilateral?

1. Square

2. Rectangle

3. Parallelogram

4. Trapezium

Q.

In a cyclic quadrilateral ABCD, the diagonal AC bisects the $\angle BCD$. Prove that the diagonal BD is parallel to the tangent to the circle at point A.

Q.

If the sum of a pair of opposite angles of a quadrilateral is ${180}^{\circ }$, then the quadrilateral is

1. Parabolic

2. Cyclic

3. Both cyclic and parabolic

4. Can't say

Q.

Two circles intersect each other at points A and B. A straight line PAQ cuts the circles at P and Q. If the tangent at P and Q intersect at point T; show that the points P, B, Q and T are concyclic.

Q. Question 13
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle.

Q.

ABCD is a cyclic quadrilateral. Then which of the angles given add to ${180}^{\circ }$?

1. 1

2. 2

3. 3
4. 4

Q.

In the figure, given below, AD = BC,

$\angle BAC={30}^{\circ }$ and $\angle CBD={70}^{\circ }$. Find: $\angle ADB$

1. ${10}^{\circ }$

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

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

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

Q. If A, B, C, D are angles of a cyclic quadrilateral , find the value of $cosA+cosB+cosC+cosD$.

1. $0$
2. $1$
3. $2$
4. $-1$

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 passing through the centre O. Then, find the value of x.

Q. If the given figure, $AB,BC$ and $CD$ are equal chords of a circle with centre $O$ and $AD$ is a diameter. If $\angle DEF={110}^o$. Then (i) $\angle AEF$ (ii) $\angle FAB$ are respectively:

1. ${20}^o\&{130}^o$
2. ${30}^o\&{130}^o$
3. ${20}^o\&{120}^o$
4. ${15}^o\&{130}^o$

Q. Question 13
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle.

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. ${60}^{\circ }$

2. ${30}^{\circ }$

3. ${120}^{\circ }$

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

Q.

ABCD is a cyclic quadrilateral BA and CD produced, meet at E . Prove that $\Delta EBC$ and $\Delta EDA$ are equiangular.

Q. To draw a pair of tangents to a circle which are inclined to each other at an angle of ${75}^{\circ }$. It is required to draw tangents at the endpoints of two radii of the circle, the angle between them should be.

1. ${105}^{\circ }$
2. ${65}^{\circ }$
3. ${95}^{\circ }$
4. ${75}^{\circ }$

Q. Find the four angles of a cyclic quadrilateral ABCD in which $\angle A=(2x-1{)}^o,\angle B=(y+5{)}^o,\angle C=(2y+15{)}^o$ and $\angle D(4x-7{)}^o$

1. $\angle A={35}^o,\angle B={75}^o,\angle C={95}^o,\angle D={135}^o$
2. $\angle A={65}^o,\angle B={55}^o,\angle C={115}^o,\angle D={125}^o$
3. $\angle A={25}^o,\angle B={45}^o,\angle C={105}^o,\angle D={135}^o$
4. $\angle A={45}^o,\angle B={75}^o,\angle C={95}^o,\angle D={125}^o$

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, find the value of x.

Q.
If $\angle BCD={50}^{\circ },$ then $\angle BAD$ = ___.

1. ${50}^{\circ }$
2. ${100}^{\circ }$
3. ${25}^{\circ }$
4. ${70}^{\circ }$

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, O is the centre of the circle $\angle$OAB = 30${}^o$ and $\angle$OCB = 40${}^o$. Then $\angle$AOC is

1. ${70}^o$
2. ${20}^o$
3. ${60}^o$
4. ${120}^o$

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. ${60}^{\circ }$

2. ${30}^{\circ }$

3. ${120}^{\circ }$

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

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. ${60}^{\circ }$

2. ${30}^{\circ }$

3. ${120}^{\circ }$

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

Q. Question 6
To draw a pair of tangents to a circle which are inclined to each other at an angle of ${60}^{\circ }$, it is required to draw tangents at endpoints of those two radii of the circle, the angle between them should be
(A) ${135}^{\circ }$
(B) ${90}^{\circ }$
(C) ${60}^{\circ }$
(D) ${120}^{\circ }$

Q.
A tangent PA is drawn from an external point P to a circle of radius $3\sqrt{2}$ cm such that the distance of the point P from O is 6 cm as shown in the figure. The value of $\angle APO$ is

1. ${30}^{\circ }$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. ${75}^{\circ }$

Q.

If a pair of opposite sides of a cylic quadrilateral are equal, prove that its diagonals are equal. [3 MARKS]

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. Statement 1 and statement 5 are true

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

3. Only statement 4 is true

4. Only statement 1 is true