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
- 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
Mensuration
Areas - Heron’S Formula
Surface Areas and Volumes
Statistics and Probability
Statistics
Probability
text
Axiom : If a ray stands on a line, then the sum of two adjacent angles so formed is 180°.
The sum of two adjacent angles is 180°, then they are called a linear pair of angles. It is given that ‘a ray stands on a line’. From this ‘given’, we have concluded that ‘the sum of two adjacent angles so formed is 180°’.
The ‘conclusion’ of Axiom as ‘given’ and the ‘given’ as the ‘conclusion’. So it becomes:
(A) If the sum of two adjacent angles is 180°, then a ray stands on a line (that is, the non-common arms form a line).
The Axiom and statement (A) are in a sense the reverse of each others. We call each as converse of the other.
Axiom: If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line. For obvious reasons, the two axioms above together is called the Linear Pair Axiom. Two lines intersect, the vertically opposite angles are equal.
theorem
Theorem: If two lines intersect each other, then the vertically opposite angles are equal.
Proof : In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 
They lead to two pairs of vertically opposite angles, namely,
(i) ∠ AOC and ∠ BOD (ii) ∠ AOD and ∠ BOC.
We need to prove that ∠ AOC = ∠ BOD and ∠ AOD = ∠ BOC.
Now, ray OA stands on line CD.
Therefore, ∠ AOC + ∠ AOD = 180° (Linear pair axiom) (1)
Can we write ∠ AOD + ∠ BOD = 180° (2)
From (1) and (2), we can write
∠ AOC + ∠ AOD = ∠ AOD + ∠ BOD
This implies that ∠ AOC = ∠ BOD
Similarly, it can be proved that ∠AOD = ∠BOC.
