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°
