NCERT CBSE Class 9th Mathematics geometry : Introduction to Euclid's Geometry

THEORETICAL GEOMETRY

The word “geometry” is derived from the combination of two Greek words “geo” and “metron”. The word “geo” means “earth” and “metron” means “measurement”. Thus the subject of “earth measurement” was originally named as “geometry”.

Basic Geometrical terms

Point: A point is used to represent a position in space. In practice, we put a small dot on a paper or on a black board to indicate a point. But theoretically, a point has no size or shape. A point can also be understood as the position where two lines intersect each other.

Line: a line ise the set of points lying at the intersection of two planes.

Plane : a plane is  a surface extending infinitely in all directions such that all points lying on the line joining any two points on the surface lie on the surface itself.

Points are denoted by capital letters such as A, B, C and D. If A and B are two points on a line, then the line is denoted by writing↔AB and is read as ‘the line AB’.

The line extends infinitely in two directions

Line segment : If A and B are two points on a line, then the portion of the line between A and B, including the points A and B, is called the line segment between A and B and is denoted by the symbol AB. The length of AB is denoted simply by writing AB

Ray: A ray is the portion of a line starting from a point on the line extending in one of the two directions of the line. The starting point is called the initial point of the ray.

Two rays →AB and having the common initial point A is said to form an angle at A. The point A is called the vertex of the angle. We shall denote the angle as <BAC or<CAB . Here the segments AB and AC are called a pair of arms of the angle. 

Angles are measured by a unit called degree.

Supplementary angles If the sum of the measures of two adjacent angles is 180º, then the two angles are called supplementary angles

Axioms and Theorems on lines

Property 1: Given any two points on a plane, there is one and only one line containing them. From the above property, we observe that

(i) Two distinct points in a plane determine a unique line. If X and Y are any two points on a line, the line XY (↔)is denoted simply as the line XY, omitting ‘↔’.

(ii) Three or more points are called collinear points if they all lie on the same line.

(iii) Three or more points are called non-collinear points if at least one of them does not lie on the line passing through two of the points.

Property 2: Two distinct lines cannot have more than one point in common.

If two distinct lines have a common point, then the lines are called intersecting lines.

If two distinct lines in a plane have no point in common, then the two lines are called non-intersecting lines. 

Two non-intersecting lines are called parallel lines.

Property 3: Given a line and a point not on it, there is one and only one line that passes through the given 

point and is parallel to the given line.

If three or more lines pass through the same point, then the lines are called concurrent lines.

Property 4: If two lines intersect, then the vertically opposite angles are equal.

Property 5: If a transversal intersects two parallel lines, then any pair of corresponding angles are equal

Theorem 1: If a transversal intersects two parallel lines, then any pair of alternate interior angles are equal.

Theorem 2: If a transversal intersects two parallel lines, then any pair of alternate (interior or exterior) angles 

are equal.

Theorem 3: If a transversal intersects two parallel lines, then any pair of interior angles are supplementary.

Property 6: If a transversal cuts two lines such that a pair of alternate angles are equal, then the lines are 

parallel.

Theorem 4: If a transversal intersects two lines such that a pair of corresponding angles are equal, then the 

lines are parallel.

Theorem 5: If a transversal cuts two lines such that a pair of interior angles on the same side of the 

transversal are supplementary, then the lines are parallel.

Theorem 7: If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the 

interior opposite angles.

Property 7: The sum of any two sides of a triangle is greater than the third side.( Note: We observe that in 

any triangle ABC,

(i) AB < BC + CA (ii) BC < CA + AB (iii) CA < AB + BC.

These inequalities are called triangle inequalities.)

Property 8: In any triangle, the largest side has the greatest angle opposite to it.

Property 9: If any two sides and the included angle of one triangle are equal to any two sides and the 

included angle of another triangle, then the two triangles are congruent.

Note: The above property is known as Side-Angle-Side criterion or simply SAS criterion for congruence of 

triangles.

Theorem 8: Two triangles are congruent if any two angles and the included side of one triangle are 

equal to the two angles and the included side of the other triangle.

Note: The above criterion is known as Angle-Side-Angle criterion or simply as ASA criterion. We observe that in this criterion, a side and the angles on this side of one triangle should correspond to a side and the angles on it of another triangle for congruency.

Theorem 9: Two triangles are congruent if any two angles and a side of one triangle are equal to the two angles and the corresponding side of the other triangle.

Note: The above criterion for congruency is known as the Angle-Angle-Side or AAS Criterion.

Theorem 11: Two right triangles are congruent if the hypotenuse and a side of one triangle are respectively equal to the hypotenuse and a side of the other triangle.

Note: The above criterion is known as the Right-Hypotenuse-Side or RHS criterion for congruence of right triangles.