Converse of Cyclic Quadrilateral Theorem

Q.

What are the properties of a right-angle triangle?

Q. In the given figure, ABCD is a cyclic quadrilateral in which AC and BD are its diagonals. If $\angle DBC={55}^{\circ }$ and $\angle BAC={45}^{\circ }$, then $\angle BCD$ = ___ .

1. ${80}^{\circ }$
2. ${70}^{\circ }$
3. ${60}^{\circ }$
4. ${90}^{\circ }$

Q. Question 17
A quadrilateral ABCD is inscribed in a circle such that AB is a diameter and $\angle ADC={130}^{\circ }$ . Find $\angle BAC$.

Q. In the cyclic quadrilateral WXYZ on the circle centred at O, $\angle ZYW={10}^{\circ }$ and $\angle YOW={100}^{\circ }$. What is the measure of $\angle YWZ$?

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

Q. Question 6
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 $\angle ECD$.

Q. In the given figure, ABCD is a cyclic quadrilateral. What is the value of $\frac{x^{\circ }}{{50}^{\circ }}$?

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

Q.

In the given figure, $\angle A={70}^{\circ }$ and $\angle ABC={50}^{\circ }$ , find $\angle DPC$ and $\angle BQC$.

1. 30^o

2. 40^o

3. 20^o

4. 50^o

Q.

\(\text{(D) In the given figure, O is the centre of the arc ABC which subtends an angle of} 130^\circ~ \text{at the centre.}\text{If AB is extended to P then}~\angle PBC = ___\)

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

2. ${130}^{\circ }$

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

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

Q.

Match the two column;
$\begin{array}{cc}\text{Column-I}& \text{Column-II}\\ \text{(A) In the given figure,}& \text{(p)}{60}^{\circ }\\ \text{ABCD is a cyclic Quadrilateral.}& \\ \text{O is the centre of the circle.}& \\ \text{If}\angle \text{BOD}={160}^{\circ }\text{find the Measure of}\angle \text{BPD.}& \end{array}$


$\begin{array}{cc}\text{(B) In given figure, ABCD is a cyclic}& \text{(q)}{65}^{\circ }\\ \text{quadrilateral whose side AB is a diameter}& \\ \text{of the circle through A, B, C, D.}& \\ \text{If}\angle {\text{ADC = 130}}^{\circ }\text{, find}\angle \text{BAC.}& \end{array}$


$\begin{array}{cc}\text{(C) In the given figure BD = DC and}& {\text{(r) 40}}^{\circ }\\ \angle {\text{CBD = 30}}^{\circ }\text{Find m}\left(\angle \text{BAC}\right)& \end{array}$


$\begin{array}{cc}\text{(D) In the given figure, O is the centre of the}& {\text{(s) 100}}^{\circ }\\ {\text{arc ABC subtends an angle of 130}}^{\circ }\text{at the centre.}& \\ \text{If AB is extended to Pm find}\angle \text{PBC.}& \end{array}$


Choose the correct option:

1. A-s; B-r; C-p; D-q

2. A-s; B-p; C-q; D-r

3. A-p; B-q; C-r; D-s

4. A-s; B-p; C-r; D-q

Q. In the figure at right, ABCD is a cyclic quadrilateral whose diagonals intersect at P such that $\angle DBC={30}^o$ and $\angle BAC={50}^o$.
Find (i) $\angle BCD$ (ii) $\angle CAD$

Q. Prove that the angle subtend in a semi circle is always ${90}^o$

Q.

If the arms of one angle are respectively parallel to the arms of another angle, then the two angles are :

1. Neither equal nor supplementary

2. Not equal but supplementary

3. Equal but not supplementary

4. Either equal or supplementary.

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

Q. In the given figure, the points P, Q, R, S, and T lie on the circle. Prove that △OST is an isosceles triangle. [2 Marks]

Q. In the given figure, $PQ$ is a tangent to a circle at $A,DB$ is the diameter and $O$ is the centre of the circle. If $\angle ADB={30}^{o}and\angle CBD={60}^o$, calculate:
i) $\angle QAB$
ii) $\angle PAD$
iii) $\angle CDB$

Q. ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If $\angle DBC={80}^{\circ },\angle BAC$ is ${40}^{\circ },$ find $\angle BCD$.

1. ${50}^{\circ }$
2. ${40}^{\circ }$
3. ${60}^{\circ }$
4. ${80}^{\circ }$

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

1. ${115}^{\circ }$
2. ${72}^{\circ }$
3. ${88}^{\circ }$
4. ${108}^{\circ }$

Q. In the given figure $PQRS$ is a cyclic quadrilateral $PQ$ and $SR$ produced meet at $T$.
Prove $△TPS\sim △TRQ$

Q.

$BC$ is a chord of a circle with centre $O$.A is a point on major are $BC$.Find the total measure of $\angle BACand\angle OBC.$

1. $90°$

2. $100°$

3. $120°$

4. $150°$

Q. In the figure (ii) given below, AB is the diameter of the semicircle ABCDE with center O. If AE=ED and $\angle BCD={140}^{\circ }$, find $\angle AED$ and $\angle EBD$. Also prove that OE is parallel to BD.

Q.

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

1. trapezium

2. cyclic quadrilateral

3. kite

4. parallelogram

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.

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If $\angle DBC={80}^{\circ }$ and $\angle BAC={40}^{\circ },$ then find $\angle BCD$.

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

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

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

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

Q. $ABC$ is an isosceles triangle in the given circle with centre $O,$ if $\angle ABC={42}^{\circ },$ then find the measure of $\angle CDE.$

1. ${84}^{\circ }$
2. ${138}^{\circ }$
3. ${96}^{\circ }$
4. ${148}^{\circ }$

Q. In Fig., $\angle BAD={78}^o,\angle DCF=x^o$ and $\angle DEF=y^o$. Find the values of $x$ and $y$

Q. In the figure, $ABCD$ is a cyclic quadrilateral with $BC=CD$.