Theorem on Cyclic Quadrilateral and Its Converse

Q. If $PQRS$ is a cyclic quadrilateral in which $\angle PSR={70}^{\circ }$ and $\angle QPR={40}^{\circ }$, then find $\angle PRQ$.

Q.

Find x, y and z in the figure given below.

1. $x={70}^{\circ },y={110}^{\circ },z={70}^{\circ }$

2. $x={110}^{\circ },y={110}^{\circ },z={70}^{\circ }$

3. $x={110}^{\circ },y={110}^{\circ },z={90}^{\circ }$

4. $x={110}^{\circ },y={110}^{\circ },z={110}^{\circ }$

Q.
Find the value of 𝑥 in the given figure.

1. ${50}^{\circ }$
2. ${55}^{\circ }$
3. ${65}^{\circ }$
4. ${60}^{\circ }$

Q. PQRS is a cyclic quadrilateral. If $\angle Q=\angle R={65}^{\circ },$ then find $\angle P$= ___ and $\angle S$ = __.

1. ${120}^{\circ },{110}^{\circ }$
2. ${115}^{\circ },{115}^{\circ }$
3. ${125}^{\circ },{115}^{\circ }$
4. ${115}^{\circ },{125}^{\circ }$

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

1. ${25}^{\circ }$
2. ${60}^{\circ }$
3. ${50}^{\circ }$
4. ${40}^{\circ }$

Q.

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

1. 90°

2. 180°

3. 360°

4. 45°

Q. In the given figure, A, B and C are three points on the circle with centre O such that $\angle BOC={30}^{\circ }$ and $\angle AOB={60}^{\circ }$. If D is a point on the circle that is not on the arc ABC, then find $\angle ABC.$

1. ${135}^{\circ }$
2. ${45}^{\circ }$
3. ${120}^{\circ }$
4. $(\frac{60}{2}{)}^{\circ }$

Q. In the given figure, what is the value of $x$ ?

1. ${30}^{\circ }$
2. ${20}^{\circ }$
3. ${15}^{\circ }$
4. ${10}^{\circ }$

Q. In the given figure, ABCD is a cyclic quadrilateral, AE is drawnn parallel to CD and BA is produced. If $\angle ABC={92}^{\circ }$ and $\angle FAE={20}^{\circ }$, then $\angle BCD$ is equal to

Q. In the given figure , AE is the diameter of the circle. Write down the numerical value of $\angle ABC+\angle CDE$.

1. ${180}^{\circ }$
2. ${270}^{\circ }$
3. ${360}^{\circ }$
4. ${540}^{\circ }$

Q.

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

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

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

3. ${360}^{\circ }$

4. ${45}^{\circ }$

Q.

In the given figure, if $\angle AOC={130}^{\circ },$ then the value of $\angle ABC$ = _______.

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

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

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

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