#### Topics

##### Number Systems

##### Number Systems

##### Algebra

##### Polynomials

##### Linear Equations in Two Variables

##### Coordinate Geometry

##### Geometry

##### Coordinate Geometry

##### Mensuration

##### Introduction to Euclid’S Geometry

##### Lines and Angles

- Introduction to Lines and Angles
- Basic Terms and Definitions
- Intersecting Lines and Non-intersecting Lines
- Parallel Lines
- Pairs of Angles
- Parallel Lines and a Transversal
- Lines Parallel to the Same Line
- Angle Sum Property of a Triangle

##### Statistics and Probability

##### Triangles

##### Quadrilaterals

- Concept of Quadrilaterals - Sides, Adjacent Sides, Opposite Sides, Angle, Adjacent Angles and Opposite Angles
- Angle Sum Property of a Quadrilateral
- Types of Quadrilaterals
- Another Condition for a Quadrilateral to Be a Parallelogram
- Theorem of Midpoints of Two Sides of a Triangle
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles.
- Theorem : If Each Pair of Opposite Sides of a Quadrilateral is Equal, Then It is a Parallelogram.
- Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Theorem: If in a Quadrilateral, Each Pair of Opposite Angles is Equal, Then It is a Parallelogram.
- Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
- Theorem : If the Diagonals of a Quadrilateral Bisect Each Other, Then It is a Parallelogram

##### Circles

- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- Angle Subtended by a Chord at a Point
- Perpendicular from the Centre to a Chord
- Circles Passing Through One, Two, Three Points
- Equal Chords and Their Distances from the Centre
- Angle Subtended by an Arc of a Circle
- Cyclic Quadrilateral

##### Areas - Heron’S Formula

##### Surface Areas and Volumes

##### Statistics

##### Algebraic Expressions

##### Algebraic Identities

##### Area

##### Constructions

- Introduction of Constructions
- Basic Constructions
- Some Constructions of Triangles

##### Probability

## Notes

Take a line segment PQ and a point R not on the line containing PQ. Join PR and QR (see following Fig.).

Then ∠ PRQ is called the angle subtended by the line segment PQ at the point R. What are angles POQ, PRQ and PSQ called in following Fig.

∠ POQ is the angle subtended by the chord PQ at the centre O,

∠ PRQ and ∠ PSQ are respectively the angles subtended by PQ at points R and S on the major and minor arcs PQ.

There different chords of a circle and angles subtended by them at the centre that the longer is the chord, the bigger will be the angle subtended by it at the centre.

Draw two or more equal chords of a circle and measure the angles subtended by them at the centre.

## Theorem

**Theorem:** Equal chords of a circle subtend equal angles at the centre.**Proof :** You are given two equal chords AB and CD of a circle with centre O (see following Fig.). You want to prove that ∠ AOB = ∠ COD.

In triangles AOB and COD,

OA = OC (Radii of a circle)

OB = OD (Radii of a circle)

AB = CD (Given) Therefore, ∆ AOB ≅ ∆ COD (SSS rule) This gives ∠ AOB = ∠ COD

(Corresponding parts of congruent triangles)

**Theorem:** If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.

**Given :** ∠AOB = ∠ POQ**To prove :** AB = PQ**Proof:** In triangles AOB and POQ,

∠AOB = ∠POQ (Given)

OA =OP (Radii of same circle)

OB = OQ (Radii of same circle)

∆ AOB ≅ ∆ POQ (SAS congruence rule)

AB = PQ (CPCT)

Hence , the theorem is proved.