#### 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

**Sum of Four Angles of a Quadrilateral:**

- Cut out a paper in the shape of a quadrilateral.
- Make folds in it that join the vertices of opposite angles.

- Take two triangular pieces of paper such that one side of one triangle is equal to one side of the other.
- Let us suppose that in ∆ABC and ∆PQR, sides AC and PQ are the equal sides.

- Join the triangles so that their equal sides lie side by side.
- We used two triangles to obtain a quadrilateral. The sum of the three angles of a triangle is 180°.
- Hence, The sum of the measures of the four angles of a quadrilateral is 360°.

## Theorem

**Angle Sum Property of a Quadrilateral:**

**Theorem:** The sum of the angles of a quadrilateral is 360°.

**Construction:** This can be verified by drawing a diagonal AC and dividing the quadrilateral into two triangles.

**Proof:**

Let ABCD be a quadrilateral and AC be diagonal.

In △ ABC,

You know that,

∠ B + ∠ BAC + ∠ BCA = 180°........(1)

Similarly, in △ADC,

∠ D + ∠ DAC + ∠ DCA = 180°........(2)

Adding (1) and (2), we get,

∠ B + ∠ BAC + ∠ BCA + ∠ D + ∠ DAC + ∠ DCA = 180° + 180°

Also, ∠ BAC + ∠ DAC = ∠ A and ∠ BCA + ∠ DCA = ∠ C

So, ∠ A + ∠ B + ∠ C + ∠ D = 180° + 180°= 360°

i.e., The sum of the angles of a quadrilateral is 360°.

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