Angles in Same Segment of a Circle

Q.

AB and CD are two equal chords of a circle with centre O such that $\angle AOB={80}^{\circ }$. Then, $\angle COD$ = ?

(a) ${100}^{\circ }$

(b) ${80}^{\circ }$

(c) ${120}^{\circ }$

(d) ${40}^{\circ }$

Q.

Prove that opposite angles of a cyclic quadrilateral are supplementary:

Q.

In the figure, if chords AB and CD of the circle intersect each other at right angles, then $x+y=$

1. ${60}^o$

2. ${45}^o$

3. ${75}^o$

4. ${90}^o$

Q.

In the figure, If $\angle ACB={40}^{\circ }.\angle DPB={120}^{\circ },$ find $\angle CBD$.

Q.

In the figure, O is the centre of the circle. If $\angle BOD={160}^{\circ }$, find the values of x and y.

Q.

In the figure, AB and CD are diameters of a circle with centre O. If $\angle OBD={50}^{\circ }$, find $\angle AOC$.

Q. Question 9
Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see the given figure). Prove that $\angle ACP=\angle QCD$

Q. Question 8
If the length of an arc of a circle of radius r is equal to that of an arc of a circle of radius 2r , then the angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle is this statement false? Why?

Q.

In the given figure, $AB$ and $CD$ are straight lines through the centre $O$ of a circle. If $\angle AOC={80}^{\circ }and\angle CDE={40}^{\circ }$ , find (i) $\angle DCE,\left(ii\right)\angle ABC.$

Q.

In the given figure, $△$ABC is equilateral. Find $\left(i\right)\angle BDC,\left(ii\right)\angle BEC.$

Q. Question 10
In any triangle ABC, if the angle bisector of $\angle A$ and perpendicular bisector of BC intersect, prove that they intersect on the circum circle of the triangle ABC.

Q.

In the figure, AB is a diameter of the circle such that $\angle A={35}^{o}and\angle Q={25}^o$, find $\angle PBR$.

Q.

In the figure, O is the centre of the circle, prove that $\angle x=\angle y+\angle z$.

Q.

Bisectors of angles $\angle A,\angle B$ and $\angle C$ of a triangle $ABC$ intersect its circumcircle at $D,E$ and $F$ respectively. Prove that the angles of the triangle $DEF$ are
${90}^{\circ }-\frac{1}{2}\angle A,{90}^{\circ }-\frac{1}{2}\angle B$ and ${90}^{\circ }-\frac{1}{2}\angle C$

Q.

Question 9
Two chords AB and AC of a circle subtends angles equal to ${90}^{\circ }and{150}^{\circ }$ respectively at the centre. Find $\angle BAC$, if AB and AC lie on the opposite sides of the centre.

Q.

In the given figure, AB and CD are two intersecting chords of a circle. If $\angle CAB={40}^{\circ }and\angle BCD={80}^{\circ }\text{then}\angle CBD=?$

(a) ${80}^{\circ }$

(b) ${60}^{\circ }$

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

(d) ${70}^{\circ }$

Q.

In the given figure, O is the centre of the circle. $\angle \text{PBC}={25}^{\circ }\text{and}\angle \text{APB}={110}^{\circ }$, find the value of $\angle \text{ADB.}$

Q. In the given figure, ΔABC is isosceles triangle with $AB=ACand\angle ABC={40}^{\circ }$. The difference of $\angle BDCand\angle BEC$ is equal to

1. 
   ${35}^{\circ }$
2. 
   ${20}^{\circ }$
3. 
   ${30}^{\circ }$
4. 
   ${25}^{\circ }$

Q.

Angles in the same segment of a circle are

(a) equal

(b) complementary

(c) supplementary

(d) none of these

Q. Question 20
In figure, $\angle OAB={30}^{\circ }and\angle OCB={57}^{\circ }.Find\angle BOCand\angle AOC$.

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. ${40}^{\circ }$

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

Q.

In the figure, ABC is an equilateral triangle. Find $\angle BDCand\angle BEC$.

Q. In figure AB || CD. Determine $\angle a$

Q.

Question 9
Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see the given figure). Prove that $\angle ACP=\angle QCD$

Q.

Question 5
In the given figure, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that $\angle BEC={130}^{\circ }$ and $\angle ECD={20}^{\circ }$. Find $\angle BAC$.

Q. The angle subtended by a major arc in the alternate segment is obtuse.

1. True
2. False

Q. Question 1
In the given figure, sides QP and RQ of $\Delta PQR$ are produced to point S and T respectively, If $\angle SPR={135}^{\circ }and\angle PQT={110}^{\circ }$, find $\angle PRQ.$

Q. If the supplement of an angle is seven times its complement. Find the complement of the angle.

Q. Find the unknown angles x and y in the given figure, where O is the centre of the circle.

Q. Find angle BOC in the given figure if AOB = 30 degrees and AC is the diameter of the given circle.