#### Topics

##### Number Systems

##### Number Systems

##### Algebra

##### Polynomials

##### Linear Equations in Two Variables

##### Algebraic Expressions

##### Algebraic Identities

##### Coordinate Geometry

##### Geometry

##### Introduction to Euclid’S Geometry

##### Lines and Angles

##### 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
- Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles.
- Another Condition for a Quadrilateral to Be a Parallelogram
- The Mid-point Theorem
- Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles.
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- 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

##### Area

##### 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

##### Constructions

##### Mensuration

##### Areas - Heron’S Formula

##### Surface Areas and Volumes

##### Statistics and Probability

##### Statistics

##### Probability

#### theorem

**Theorem:** If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.**Proof:** In quadrilateral ABCD

∠A = ∠D

`=>` ∠A +∠B +∠C + ∠D

Now, ∠A +∠B +∠C + ∠D = 360° (angle sum property of quadrilateral)

`=>` 2(∠A +∠B ) = 360°

`=> ` ∠A +∠B = 180°

`therefore` ∠A +∠B = ∠C + ∠D = 180°

Line AB intersects AD and BC at A and B respectively.

Such that ∠A +∠B = 180°

`therefore` AD || BC (Sum of consecutive interior angle is 180° ) ...(1)

∠A +∠B = 180°

∠A +∠D = 180° (∠B= ∠D)

`therefore` AB || CD ..(2)

From (1) and (2), we get

AB || CD and AD || BC

`therefore` ABCD is a parallelogram.

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