Angles in Same Segment of a Circle

Q.

If two sides of a cyclic-quadrilateral are parallel; prove that :

(i) its other two sides are equal.

(ii) its diagonals are equal.

Q.

In the given figure, $AB=AC=CD$ and $\angle ADC={38}^{\circ }$.

Calculate :

(i) $\angle ABC$

(ii) $\angle BEC$

Q. Question 10
Two lines l and m are perpendicular to the same line n. Are l and m perpendicular to each other? Give the reason for your answer.

Q.

In $\Delta ABC$ and $\Delta DEF$, $\angle A=\angle E={40}^{\circ }$, AB : ED = AC : EF and $\angle F={65}^{\circ }$, then the value of $\angle B$ is

1. ${85}^{\circ }$

2. ${35}^{\circ }$

3. ${65}^{\circ }$

4. ${75}^{\circ }$

Q.

In the given figure, PQ = PR and $\angle PRQ={70}^{\circ }$. Find $\angle QAR$.

1. ${40}^{\circ }$

2. ${140}^{\circ }$

3. ${80}^{\circ }$

4. ${70}^{\circ }$

Q.

Question 47
The angles P, Q, R and S of a quadrilateral are in the ratio 1 : 3 : 7 : 9. Then, PQRS is a
a) Parallelogram
b) Trapezium with $PQ\parallel RS$
c) Trapezium with $QR\parallel PS$
d) Kite

Q.

Chords AB and CD of a circle with centre O intersect at a point E. If OE bisects angle AED, then prove that AB = CD.

Q.

In the given figure, AB is a diameter of the circle with centre O. DO is parallel to CB and $\angle$ DCB = ${120}^o$. Calculate :

(i) $\angle$ DAB, (ii) $\angle$ DBA,

(iii) $\angle$ DBC, (iii) $\angle$ ADC.

Also, show that the $\Delta$ AOD is an equilateral triangle.

Q.

If I is the incentre of triangle ABC and AI when produced meets the circumcircle of triangle ABC in point D.

If $\angle BAC={66}^{\circ }$ and $\angle ABC={80}^{\circ }.$

Calculate:

(i) $\angle DBC$ (ii) $\angle IBC,$ (iii) $\angle BIC.$

Q.

In figure, arc AB is congruent to arc AC and O is the centre of the circle. Prove that OA is the perpendicular bisector of BC. [4 MARKS]

Q.

In the figure, given alongside, CP bisects angle ACB.

Show that DP bisects angle ADB.

Q.

In the figure, $\angle DBC={58}^{\circ }.$ BD is a diameter of the circle. Calculate :

(i) $\angle BDC$

(ii) $\angle BEC$

(iii) $\angle BAC$

Q.

In the given figure, O is the centre of the circle $\angle PBA={45}^{\circ }$. The value of $\angle PQB$ is

1. ${55}^{\circ }$

2. ${45}^{\circ }$

3. ${65}^{\circ }$

4. ${75}^{\circ }$

Q.

In the given figure, M is the centre of the circle. Chords AB and CD are perpendiculat to each other.

If $\angle$ MAD = x and $\angle$ BAC = y :

(i) express $\angle$ AMD in terms of x.

(ii) express $\angle$ ABD in terms of y.

(iii) prove that : x = y.

Q.

In the following figure, O is centre of the circle and $\Delta ABC$ is equilateral. Which of the following is true?

1. $\angle ACB={130}^{\circ }$

2. $\angle ACB={60}^{\circ }$

3. $\angle AEB={120}^{\circ }$

4. $\angle AEB={220}^{\circ }$

Q.

In the given figure, AB is parallel to DC. $BCE={80}^{\circ }and\angle BAC={25}^{\circ }.$

Find :

(i) $\angle CAD$ (ii) $CBD$ (iii) $ADC$

Q.

In the following figure, ΔODC ∼ ΔOBA, ∠BOC = 125° and ∠CDO = 70°. Find ∠DOC, ∠DCO and ∠OAB

Q. Find QP using given information in the figure.

Q.

Equal chords AB and CD of a circle with centre O, cut at right angles at E. If M and N are the midpoints of AB and CD respectively, prove that OMEN is a square.
[4 MARKS]

Q.

In the given circle with diameter AB, find the value of x.

Q.

In triangle PQR, ST is a line intersecting PQ and PR in S and T respectively such that ST ||QR. If PS=x, SQ=2, PT=x+3, TR=x-3.What is the value of x?

1. 8

2. 6

3. 4

4. 9

Q.

ABCDE is a cyclic pentagon with centre of its circumcircle at point O such that AB = BC = CD and angle ABC = ${120}^o$.

Calculate :

(i) $\angle$ BEC (ii) $\angle$ BED

Q.

The given figure shown a circle with centre O and $\angle$ ABP = ${42}^o$.

Calculate the measure of :

(i) $\angle$ PQB

(ii) $\angle$ QPB + $\angle$ PBQ

Q.

In the figure, given below, AD = BC, $\angle BAC={30}^{\circ }and\angle CBD={70}^{\circ }.$

Find :

(i) $\angle BCD$

(ii) $\angle BCA$

(iii) $\angle ABC$

(iv) $\angle ADB$

Q.

The given figure shows a circle through the points A, B, C, and D. If $\angle BAC={67}^{\circ }$, find $\angle DBC+\angle DCB$

1. ${128}^{\circ }$

2. ${134}^{\circ }$

3. ${113}^{\circ }$

4. ${67}^{\circ }$

Q. Question 2
If one angle of a triangle is equal to the sum of the other two angles, then the triangle is:

A) An isosceles triangle
B) An obtuse triangle
C) An equilateral triangle
​​​​​​​D) ​​​​​​​A right triangle

Q.

In the given figure, AOC is a diameter and AC is parallel ot ED. If $\angle$ CBE = ${64}^o$, calculate $\angle$ DEC.

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

Q.

AB is a diameter of the circle APBR as shown in the figure. APQ and RBQ are straight lines. Select the statements that are true.

1. $\therefore \angle PRB={35}^{\circ }$

2. $\therefore \angle BPQ={90}^{\circ }$

3. $\angle BPR={30}^{\circ }$

4. $\angle PBR={115}^o$

Q. Express cos 75° + cot 75° in terms of angles between 0° and 30°.