# Angles in Alternate Segments

## Trending Questions

**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√3 cm2 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∘

b) 144∘

c) 36∘

d) 18∘

**Q.**

In the following figure, PQ and PR are tangents to the circle, with centre O. If ∠QPR=60o, calculate :

(i)∠QOR,

(ii)∠OQR,

(iii)∠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. ∠BDC = 650 Find ∠BAO.

40°

60°

30°

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 :

∠CAD=12[∠PBA−∠PAB]

**Q.**

In the given figure, △ABC is inscribed in a circle. The bisector of ∠BAC meets BC at D and the circle at E. If EC is joined, then ∠ECD=30∘. Find ∠BAC.

30°

60°

50°

70°

**Q.**

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

square

Rectangle

parallelogram

cyclic quadrilateral

**Q.**

In figure, $PA$ and $PB$ are tangents to the circle with centre $O$ such that $\xe2\u02c6APB=50\xc2\xb0$.

Write the measure of $\xe2\u02c6OAB$

**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) ∠ BAP = ∠ ADQ

(ii) ∠ AOB = 2∠ ADQ

(iii) ∠ ADQ = ∠ 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 ∠ BCG =108o 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 ∠OBC=30∘. AOC is a straight line passing through the centre O. Then, the value of x is:

60∘

30∘

120∘

90∘

**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.

45∘

55∘

60∘

70∘

**Q.**

In the given figure , △ ABC is inscribed in a circle. The bisector of ∠BAC meets BC at D and thecircle at E, if EC is joined then∠ECD=300. Find ∠BAC.

30°

60°

50°

70°

**Q.**

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

- 80∘
- 120∘
- 90∘
- 100∘

**Q.**

Question 1 (ii)

In an isosceles triangle ABC, with AB = AC, the bisectors of ∠ B and ∠ C intersect each other at O. Join A to O. Show that:

AO bisects ∠ A

**Q.**

In the above figure, ∠CAB=60∘, ∠CBA=60∘. PQ is a tangent at A. What is the value of ∠BAQ.

60∘

20∘

40∘

80∘

**Q.**

In the figure, PC is the tangent to the circle. If ∠BPC = 60∘ and ∠APB = 55∘, then find ∠ABP .

55∘

60∘

65∘

70∘

**Q.**Question 171

In a quadrilateral HOPE, PS and ES are bisectors of ∠P and ∠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 π=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.