Angles in Alternate Segments

Q.

Prove that the tangents at the extremities of any chord make equal angles with the chord. [3 MARKS]

Q.

If $△ABC$ is isosceles with AB = AC, prove that the tangent at A to the circumcircle of $△ABC$ is parallel to BC. [2 MARKS]

Q.

OABC is a rhombus whose three vertices A, B and C lie on a circle with centre O.

(i) If the radius of the circle is 10 cm, find the area of the rhombus.

(ii) If the area of the rhombus is $32\sqrt{3}c{m}^2$ find the radius of the circle.

Q.

Prove that any four vertices of a regular pentagon are concyclic (lie on the same circle).

Q. Question 26
The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. The smallest angle is
a) ${72}^{\circ }$
b) ${144}^{\circ }$
c) ${36}^{\circ }$
d) ${18}^{\circ }$

Q.

In the following figure, PQ and PR are tangents to the circle, with centre O. If $\angle QPR={60}^o$, calculate :

(i)$\angle QOR,$
(ii)$\angle OQR,$
(iii)$\angle QSR,$

Q.

A chord $PQ$ of a circle is parallel to the tangent drawn at a point $R$ of the circle. Prove that $R$ bisects the arc $PRQ$.

Q.

In the given figures, O is the centre of the circle and AB is a tangent to it at point B. $\angle$BDC = ${65}^0$ Find $\angle$BAO.

1. 40°

2. 60°

3. 30°

4. 50°

Q.

In fig., O is the center of the circle, PA and PB are tangent segments. Show that the quadrilateral AOBP is cyclic.

Q.

In teh figure; PA is a tangent to the circle, PBC is secant and AD bisects angle BAC. Show that triangle PAD is an isosceles triangle. Also, show that :

$\angle CAD=\frac{1}{2}[\angle PBA-\angle PAB]$

Q.

In the given figure, $△ABC$ is inscribed in a circle. The bisector of $\angle BAC$ meets BC at D and the circle at E. If EC is joined, then $\angle ECD={30}^{\circ }$. Find $\angle BAC$.

1. 30°

2. 60°

3. 50°

4. 70°

Q.

The quadrilateral formed by joining the angle bisectors of a cyclic quadrilateral is a

1. square

2. Rectangle

3. parallelogram

4. cyclic quadrilateral

Q.

In figure, $PA$ and $PB$ are tangents to the circle with centre $O$ such that $\angle APB=50°$.

Write the measure of $\angle OAB$

Q.

In the given figure, PAT is tangent to the circle with centre O, at point A on its circumference and is parallel to chord BC. If CDQ is a line segment, show that :

(i) $\angle$ BAP = $\angle$ ADQ

(ii) $\angle$ AOB = 2$\angle$ ADQ

(iii) $\angle$ ADQ = $\angle$ ADB.

Q.

In the figure, ABCD is a cyclic quadrilateral with BC = CD. TC is tangent to the circle at point C and DC is produced to point G. If $\angle$ BCG =1${08}^o$ and O is the centre of the circle, find :

(i) angle BCT

(ii) angle DOC

Q.

Two circles intersect each other at points A and B. Their common tangent touches the circles at points P and Q as shown in the figure. Show that the angle PAQ and PBQ are supplementary.

Q.

In the given figure, ABCD is a cyclic quadrilateral, OB is the radius, PB is the tangent at point B and $\angle OBC={30}^{\circ }$. AOC is a straight line passing through the centre O. Then, the value of x is:

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

2. ${30}^{\circ }$

3. ${120}^{\circ }$

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

Q.

Circles with centres P and Q intersect at points A and B as shown in the figure.

CBD is a line segment and EBM is tangent to the circle, with centre Q, at point B. If the circles are congruent ;

show that CE = BD.

Q.

In the figure given below, find $x$ if AB || CD.

1. 45${}^{\circ }$

2. 55${}^{\circ }$

3. 60${}^{\circ }$

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

Q.

$Inthegivenfigure,△ABCisinscribedinacircle.Thebisectorof\angle BACmeetsBCatDandthecircleatE,ifECisjoinedthen\angle ECD={30}^0.Find\angle BAC.$

1. 30°

2. 60°

3. 50°

4. 70°

Q.

If ABCD is a cyclic quadrilateral, then $x$ = ___.

1. ${80}^{\circ }$
2. ${120}^{\circ }$
3. ${90}^{\circ }$
4. ${100}^{\circ }$

Q.

Question 1 (ii)
In an isosceles triangle ABC, with AB = AC, the bisectors of $\angle Band\angle C$ intersect each other at O. Join A to O. Show that:
AO bisects $\angle A$

Q.

In the above figure, $\angle CAB={60}^{\circ },\angle CBA={60}^{\circ }$. PQ is a tangent at A. What is the value of $\angle BAQ$.

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

2. ${20}^{\circ }$

3. ${40}^{\circ }$

4. ${80}^{\circ }$

Q.

In the figure, PC is the tangent to the circle. If $\angle BPC$ = ${60}^{\circ }$ and $\angle APB$ = ${55}^{\circ }$, then find $\angle ABP$ .

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

2. ${60}^{\circ }$

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

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

Q. Question 171
In a quadrilateral HOPE, PS and ES are bisectors of $\angle P$ and $\angle E$ respectively. Give reason.

Q.

If two circles intersect at two points, then prove that their centers lie on the perpendicular bisector of the common chord. [2 MARKS]

Q.

In figure, ABCD is a trapezium with AB||DC, If $△$AED is similar to $△$BEC, prove that AD = BC

Q. Question 10
In a figure, the common tangents AB and CD to two circles with centres O and O’ intersect at E. Prove that the points O, E and O’ are collinear.

Q. In Fig., a square $OABC$ is inscribed in a quadrant $OPBQ$. If $OA=20cm$, find the area of the shaded region. (Use $\pi =3.14$)

Q. If two tangents are drawn to a circle circle from an external point, the
(i) they subtend equal angles at the centre
(ii) they are equally inclined to the segment, joining the centre to that point.