#### 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

- 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

##### 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

##### 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

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

##### Mensuration

##### Areas - Heron’S Formula

##### Surface Areas and Volumes

##### Statistics and Probability

##### Statistics

##### Probability

## Notes

**Angle Sum Property of a Triangle:**

There is a remarkable property connecting the three angles of a triangle.

- Draw a triangle. Cut on the three angles. Rearrange them as shown in the following Figure. The three angles now constitute one angle. This angle is a straight angle and so has measure 180°.

Thus, the sum of the measures of the three angles of a triangle is 180°.

∴ ∠1 + ∠2 + ∠3 = 180°

- Take a piece of paper and cut out a triangle, say, ∆ABC.

Make the altitude AM by folding ∆ABC such that it passes through A.

Fold now the three corners such that all the three vertices A, B, and C touch at M.

You find that all the three angles form together a straight angle. This again shows that the sum of the measures of the three angles of a triangle is 180°.

∴ ∠B + ∠A + ∠C = 180°

## Theorem

**Angle Sum Property of a Triangle:**

**Theorem:** The sum of the angles of a triangle is 180°.

**Construction:** Draw a line XPY parallel to QR through the opposite vertex P.

**Proof: **

In △ PQR,

Sum of all angles of a triangle is 180°.

∠PQR + ∠PRQ + ∠QPR = 180°......(1)

Since XY is a straight line, it can be concluded that:

Therefore, ∠XPY + ∠QRP + ∠RPY = 180°.

But XPY || QR and PQ, PR are transversals.

So,

∠XPY = ∠PQR.....(Pairs of alternate angles)

∠RPY = ∠PRQ.....(Pairs of alternate angles)

Substituting ∠XPY and ∠RPY in (1), we get

∠PQR + ∠PRQ + ∠QPR = 180°

Thus, The sum of the angles of a triangle is 180°.

## Example

In the given figure find m∠P.

By angle sum property of a triangle,

m∠P + 47° + 52° = 180°

Therefore,

m∠P = 180° – 47° – 52°

m∠P = 180° – 99°

m∠P = 81°