#### 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

**The diagonals of a parallelogram bisect each other.**

**Given: **ABCD is a parallelogram.

**To Prove: **AO = OC and BO = OD.

**Construction:** Draw diagonal AC and diagonal BD.

**Proof:**

ABCD is a parallelogram.

∴ AB || DC and AD || BC

Now,

AB || DC, AC is the transversal intersecting them at A and C respectively.

∠BAC = ∠DCA ........(Alternate interior angles)

Thus, ∠BAO = ∠ DCO .........(1)

As, AB || DC and BD is the transversal intersecting them at B and D respectively.

∠ABD = ∠CDB ........(Alternate interior angles)

So, ∠ABO = ∠CDO ...(2)

In △ AOB and △ COD,

∠ DCO ≅ ∠BAO .....(Pair of alternate interior angles)(1)

∠ CDO ≅ ∠ABO .....(Pair of alternate interior angles)(2)

side DC = side AB ....(parallel side)

by ASA congruency condition,

△ AOB ≅ △ COD.

This gives, AO = CO and BO = DO.....(C.P.C.T)

Hence proved.

## Example

If OE = 4 then OP also is 4. .......(The diagonals of a parallelogram bisect each other.)

So, PE = 8,

Therefore, HL = 8 + 5 = 13

Hence, OH = `1/2 xx 13` = 6.5 cms.