Sum of Pair of Opposite Angles in Quadrilateral

Q.

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

1. Parabolic

2. Cyclic

3. Both cyclic and parabolic

4. Can't say

Q.

ABCD is a cyclic quadrilateral such that $\angle ADB={30}^{o}and\angle DCA={80}^o$, then $\angle DAB=$

1. ${70}^o$

2. ${125}^o$

3. ${100}^o$

4. ${150}^o$

Q.

In the given figure, O is the centre of a circle and $\angle AOC={130}^{\circ }.Then\angle ABC=?$

(a) ${50}^{\circ }$

(b) ${65}^{\circ }$

(c) ${115}^{\circ }$

(d) ${130}^{\circ }$

Q.

In a cyclic quadrilateral ABCD, If $m\angle A=3\left(m\angle C\right)$. Find $m\angle A$.

Q.

In a cyclic quadrilateral ABCD, if $(\angle B-\angle D)={60}^{\circ }$, show that the smaller of the two is ${60}^{\circ }$.

Q.

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

Q.

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

Q.

In the figure, O is the centre of the circle and $\angle DAB={50}^o$. Calculate the values of x and y.

Q.

Prove that if an arc of a circle subtends a right angle at any point on the remaining part of the circle, then the arc is a semi-circle.

Q. Find all the angles of the given cyclic quadrilateral $ABCD$ in the figure.

Q.

In the figure, O is the centre of the circle. If $\angle CEA={30}^o$, find the values of x, y and z.

Q.

If ABCD is a cyclic quadrilateral, then which of the angles given add up to ${180}^{\circ }$?

1. $\angle$ABC and $\angle$BCD

2. $\angle$BAD and $\angle$ADC

3. $\angle$BAD and $\angle$BCD
4. $\angle$CDA and $\angle$ABD

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

Q.

C is a point on the minor arc AB of the circle, with the centre O. Given $\angle$ ACB = $x^{\circ }$ and $\angle$ AOB = $y^{\circ }$ .Express y in terms of x. Calculate x, if ACBO is a parallelogarm.

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 4 is true

2. Only statement 1 is true

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

4. Statement 1 and statement 5 are true

Q.

A quadrilateral has two of its angles as ${30}^{\circ }$ and ${40}^{\circ }$. What type of quadrilateral can it be?

1. Trapezium

2. Square

3. Parallelogram

4. Rectangle

Q.

A chord of a circle AB is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc in degree.

___

Q. $ABC$ is an equilateral triangle. Find $\angle BEC$.

1. ${30}^{\circ }$
2. ${60}^{\circ }$
3. ${90}^{\circ }$
4. ${120}^{\circ }$

Q.

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

1. 1

2. 2

3. 3
4. 4

Q. Question 9
If bisectors of opposite angles of a cyclic quadrilateral ABCD intersect the circle, circumscribing it at the point P and Q, prove that PQ is a diameter of the circle.

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

Q. Find the measure of an exterior angle at the base of an isosceles triangle whose vertex angle measures ${50}^o$.

1. ${110}^o$
2. ${100}^o$
3. ${115}^o$
4. ${65}^o$

Q. $Infigureif\angle AOB={90}^{o}and\angle ABC={30}^{o}then\angle CAOisequalto$

1. ${30}^o$
2. ${90}^o$
3. ${45}^o$
4. ${60}^o$

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. Only statement 1 is true

3. Only statement 4 is true

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

Q. In the given figure, find x+y

1. ${102}^{\circ }$
2. ${116}^{\circ }$
3. ${76}^{\circ }$
4. ${64}^{\circ }$