If in a circle AB is the chord and is making ∠ACB at any point of the circle then this is the angle subtended by the chord AB at a point C.

 Likewise, ∠AOB is the angle subtended by chord AB at point O i.e. at the centre and ∠ADB is also the angle subtended by AB at point D on the circle.

Theorem 1: Any two equal chords of a circle subtend equal angles at the centre.

Here in the circle, the two chords are given and PQ = RS with centre O.

So OP = OS = OQ = OR (all are radii of the circle)

∆POQ ≅ ∆SOR

∠POQ = ∠SOR  

This shows that the angles subtended by equal chords to the centre are also equal.

Theorem 2: If the angles made by the chords of a circle at the centre are equal, then the chords must be equal.

This theorem is the reverse of the above Theorem 1.

Perpendicular from the Centre to a Chord

Theorem 3: If we draw a perpendicular from the centre of a circle to any chord then it bisects the chord.

If we draw a perpendicular from the centre to the chord of the circle then it will bisect the chord. And the bisector will make a 90° angle to the chord.

Theorem 4: The line which is drawn from the centre of a circle to bisect a chord must be perpendicular to the chord.

If we draw a line OB from the centre of the circle O to the midpoint of the chord AC i.e. B, then OB is the perpendicular to the chord AB.

If we join OA and OC, then

In ∆OBA and ∆OBC,

AB = BC (B is the midpoint of AC)

OA = OC (Both are the radii of the same circle)

OB = OB (same side)

Hence, ΔOBA ≅ ΔOBC (both are congruent by SSS congruence rule)

⇒ ∠OBA = ∠OBC (respective angles of congruent triangles)

∠OBA + ∠OBC = ∠ABC = 180° [Linear pair]

∠OBC + ∠OBC = 180° [Since ∠OBA = ∠OBC]

2 x ∠OBC = 180°

∠OBC = 90^o

∠OBC = ∠OBA = 90°

∴ OB ⊥ AC

Circle through Three Points

Theorem 5: There is one and only one circle passing through three given non-collinear points.

In this figure, we have three non-collinear points A, B and C. Let us join AB and BC and then make the perpendicular bisector of both so that RS and PQ the perpendicular bisector of AB and BC respectively meet each other at Point O.

Now take the O as centre and OA as the radius to draw the circle which passes through the three points A, B and C.

This circle is known as Circumcircle. Its centre and radius are known as the Circumcenter and Circumradius.

Equal Chords and Their Distances from the Centre

Theorem 6: Two equal chords of a circle are at equal distance from the centre.

AB and CD are the two equal chords in the circle. If we draw the perpendicular bisector of these chords then the line segment from the centre to the chord is the distance of the chord from the centre.

If the chords are of equal size then their distance from the centre will also be equal.

Theorem 7: Chords at equal distance from the centre of a circle are also equal in length. This is the reverse of the above theorem which says that if the distance between the centre and the chords are equal then they must be of equal length.

Angle Subtended by an Arc of a Circle

The angle made by two different equal arcs to the centre of the circle will also be equal.

There are two arcs in the circle AB and CD which are equal in length.

So ∠AOB = ∠COD.

Theorem 8: The angle subtended by an arc at the centre is twice the angle subtended by the same arc at some other point on the remaining part of the circle.

In the above figure ∠POQ = 2∠PRQ.

Theorem 9: Angles from a common chord which are on the same segment of a circle are always equal.

If there are two angles subtended from a chord to any point on the circle which are on the same segment of the circle then they will be equal.

∠a = (1/2) ∠c (By theorem 8)

∠b = (1/2) ∠c

∠a = ∠b

Cyclic Quadrilaterals

If all the vertices of the quadrilateral come in a circle then it is said to be a cyclic quadrilateral.

Theorem 10: Any pair of opposite angles of a cyclic quadrilateral has the sum of 180º.

∠A + ∠B + ∠C + ∠D = 360º (angle sum property of a quadrilateral)

∠A + ∠C = 180°

∠B + ∠D = 180º

Theorem 11: If the pair of opposite angles of a quadrilateral has a sum of 180º, then the quadrilateral will be cyclic.

This is the reverse of the above theorem.

Intersecting lines: Two or more lines that have one and only one point in common.

Point of intersection: Point of intersection is a common point at which the intersecting lines meet.

Transversal: Transversal is a line that intersects two or more lines which lie in the same plane at distinct points.

Parallel lines: Parallel lines are the lines on a plane which never meet. They are at a distance apart.

Complementary Angles: Complementary angles are the angles whose total is equal to 90^o .

Supplementary Angles: Suplementary angles are the angles whose total is equal to 180^o

Adjacent Angles: Adjacent Angles are the angles which have a common vertex and a common interior points.

Linear Pair of Angles: Linear pair of angles is a pair of adjacent angles whose non-common sides are opposite rays.

Vertically Opposite Angles: Vertically opposite angles are the angles formed by two intersecting lines which have the have common arms.

Angles made by Transversal:

When two lines are intersecting by a transversal, eight angles are formed.

Transversal of Parallel Lines: If two parallel lines are intersected by a transversal, each pair of:

• Corresponding angles are congruent.
• Alternateinterior angles are congruent.
• Alternate exterior angles are congruent.

If the transversal is perpendicular to the parallellines, all of the angles formed are congruent to 90^o angles.

1. A line which intersects two or more given lines at distinct points is called a transversal to the given lines.
2. Lines in a plane are parallel if they do not intersect when produced indefinitely in either direction.
3. The distance between two intersecting lines is zero.
4. The distance between two parallel lines is the same everywhere and is equal to the perpendicular distance between them.

If two parallel lines are intersected by a transversal then:

• pairs of alternate (interior orexterior) angles are equal.
• pairs of corresponding angles are equal.
• interior angles onthe same sideof the transversal are supplementary.

6. If two non-parallel lines are intersected by transversal then none of (i), (ii) and (iii) hold true in 5. 7.

If two lines are intersected by a transversal, then they are parallel if any one of the following is true:

• The angles of a pair of corresponding angles are equal.
• The angles of a pairof alternate interior angles are equal.
• The angles of a pairof interior angles on the sameside of the transversal are supplementary.