Geometry Axioms & Theorems

 Class 9 

LINES AND ANGLES 

Axiom 1: If a rays stands on a line , then the sum of two adjacent angles so formed is 180 ⁰

Axiom 6.2 : If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line. 

Theorem 6.1 : If two lines intersect each other, then the vertically opposite angles are equal. 

Axiom 6.3 : If a transversal intersects two parallel lines, then each pair of corresponding angles is equal. 

Axiom 6.4 : If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other. 

Theorem 6.2 : If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal 

Theorem 6.3 : If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel. 

 Theorem 6.4 : If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.  

Theorem 6.5 : If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel. 

Theorem 6.6 : Lines which are parallel to the same line are parallel to each other. 

Theorem 6.7 : The sum of the angles of a triangle is 180º 

Theorem 6.8 : If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles. 

TRIANGLES: 

Axiom 7.1 (SAS congruence rule) : Two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle 

Theorem 7.1 (ASA congruence rule) : Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of other triangle 

Theorem 7.2 : Angles opposite to equal sides of an isosceles triangle are equal. 

Theorem 7.3 : The sides opposite to equal angles of a triangle are equal 

Theorem 7.4 (SSS congruence rule) : If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent. 

Theorem 7.5 (RHS congruence rule) : If in two right triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent. 

Theorem 7.6 : If two sides of a triangle are unequal, the angle opposite to the longer side is larger (or greater). 

Theorem 7.7 : In any triangle, the side opposite to the larger (greater) angle is longer. 

Theorem 7.8 : The sum of any two sides of a triangle is greater than the third side 

QUADRILATERALS 

Theorem 8.1 : A diagonal of a parallelogram divides it into two congruent triangles. 

Theorem 8.2 : In a parallelogram, opposite sides are equal. 

Theorem 8.3 : If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram. 

Theorem 8.4 : In a parallelogram, opposite angles are equal. 

Theorem 8.5 : If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram. 

Theorem 8.6 : The diagonals of a parallelogram bisect each other 

Theorem 8.7 : If the diagonals of a quadrilateral bisect each other, then it is a parallelogram 

Theorem 8.8 : A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel. 

Theorem 8.9 : The line segment joining the mid-points of two sides of a triangle is parallel to the third side. 

Theorem 8.10 : The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side.  

AREAS OF PARALLELOGRAMS AND TRIANGLES 

Theorem 9.1 : Parallelograms on the same base and between the same parallels are equal in area. 

Theorem 9.2 : Two triangles on the same base (or equal bases) and between the same parallels are  

equal in area 

Theorem 9.3 : Two triangles having the same base (or equal bases) and equal areas lie between the same parallels 

CIRCLES 

Theorem 10.1 : Equal chords of a circle subtend equal angles at the centre. 

Theorem 10.2 : If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal. 

Theorem 10.3 : The perpendicular from the centre of a circle to a chord bisects the chord. 

Theorem 10.4 : The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord. 

Theorem 10.5 : There is one and only one circle passing through three given non-collinear points. 

Theorem 10.6 : Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres). 

Theorem 10.7 : Chords equidistant from the centre of a circle are equal in length. 

Theorem 10.8 : The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. 

Theorem 10.9 : Angles in the same segment of a circle are equal 

Theorem 10.10 : If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle (i.e. they are concyclic). 

Theorem 10.11 : The sum of either pair of opposite angles of a cyclic quadrilateral is 180º.  

Theorem 10.12 : If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic. 

SURFACE AREAS AND VOLUMES 

Surface Area of a Cuboid = 2(lb + bh + hl) where l, b and h are respectively the three edges of the cuboid 

Surface Area of a Cube = 6a² 

Curved Surface Area of a Cylinder = 2πrh 

Total Surface Area of a Cylinder = 2πr(r + h) 

Curved Surface Area of a Cone

= 1/2 × l × 2πr = πrl 

L²= r² + h²

Total Surface Area of a Cone

= πrl + πr²2 = πr(l + r) 

Surface Area of a Sphere = 4 π r² 

Curved Surface Area of a Hemisphere = 2πr² 

Total Surface Area of a Hemisphere = 3πr²

Volume of a Cuboid = base area × height = length × breadth × height 

Volume of a Cube = edge × edge × edge = a³ 

Volume of a Cylinder = πr²2 

Volume of a Cone = 1/3 πr²h 

Volume of a Sphere = 4/3 3 πr³

Volume of a Hemisphere = 2/3πr³ 

ALGEBRA :

am × an = am+n

 (am)×(an) = am+n
(am)/(an) = am-n
(am)n = amn
(am)×(bm) = (ab)m
(a0)= am-m = am/am = 1
(am)×(bn) = (ab)m+n
am/bm = (a/b)m

Class 8 

UNDERSTANDING QUADRILATERALS 

Sum of the measures of the external angles of any polygon is 360°. 

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

A parallelogram is a quadrilateral whose opposite sides are parallel 

Property: The opposite sides of a parallelogram are of equal length 

Property: The opposite angles of a parallelogram are of equal measure. 

Property: The adjacent angles in a parallelogram are supplementary 

Property: The diagonals of a parallelogram bisect each other (at the point of their intersection, of course!) 

Property: The diagonals of a rhombus are perpendicular bisectors of one another 

Property: The diagonals of a rectangle are of equal length. 

Property: The diagonals of a square are perpendicular bisectors of each other 

MENSURATION  

1. Area of (i) a trapezium = half of the sum of

the lengths of parallel sides × perpendicular distance between them.

(ii) a rhombus = half the product of its diagonals. 

 2. Surface area of a solid is the sum of the areas of its faces. 

 3. Surface area of a cuboid = 2(lb + bh + hl) a cube = 6l ² a cylinder = 2πr(r + h)  

4. Amount of region occupied by a solid is called its volume.  

5. Volume of a cuboid = l × b × h a cube = l³ a cylinder = πr ²h 6.

(i) 1 cm³ = 1 mL

(ii) 1L = 1000 cm³

(iii) 1 m³ = 1000000 cm³ = 1000L 

EXPONENTS AND POWERS 

 am × an = am+n

 am / an = am-n
(am)n = amn
(am)×(bn) = (ab)m+n
(a0)= am / am = 1
am/am = (a/b)m

Class 7 

LINES AND ANGLES 

sum of the measures of two angles is 90°, the angles are called complementary angles. 

the sum of the measures of two angles is 180°, the angles are called supplementary angles. 

These angles are such that:  

(i) they have a common vertex;  

(ii) they have a common arm;  

(iii) the non-common arms are on either side of the common arm. 

 Such pairs of angles are called adjacent angles. Adjacent angles have a common vertex and a common arm but no common interior points. 

A linear pair is a pair of adjacent angles whose non-common sides are opposite rays. 

TRIANGLES 

An exterior angle of a triangle is equal to the sum of its interior opposite angles. 

Statement The total measure of the three angles of a triangle is 180⁰

A triangle in which all the three sides are of equal lengths is called an equilateral triangle. 

A triangle in which two sides are of equal lengths is called an isosceles triangle. 

1.The six elements of a triangle are its three angles and the three sides.  

2.The line segment joining a vertex of a triangle to the mid point of its opposite side is called a median of the triangle. A triangle has 3 medians.  

3.The perpendicular line segment from a vertex of a triangle to its opposite side is called an altitude of the triangle. A triangle has 3 altitudes.  

 4.An exterior angle of a triangle is formed, when a side of a triangle is produced. At each vertex, you have two ways of forming an exterior angle.  

5.A property of exterior angles: The measure of any exterior angle of a triangle is equal to the sum of the measures of its interior opposite angles.  

6.The angle sum property of a triangle: The total measure of the three angles of a triangle is 180°. 

 7. A triangle is said to be equilateral, if each one of its sides has the same length. In an equilateral triangle, each angle has measure 60° 

8. A triangle is said to be isosceles, if atleast any two of its sides are of same length. The non-equal side of an isosceles triangle is called its base; the base angles of an isosceles triangle have equal measure.  

9. Property of the lengths of sides of a triangle: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. The difference between the lengths of any two sides is smaller than the length of the third side. 

CONGRUENCE OF TRIANGLES 

If two line segments have the same (i.e., equal) length, they are congruent. Also, if two line segments are congruent, they have the same length. 

If two angles have the same measure, they are congruent. Also, if two angles are congruent, their measures are same. 

SSS Congruence Criterion:

If under a given correspondence, the three sides of one triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent. 

SAS Congruence Criterion:

If under a correspondence, two sides and the angle included between them of a triangle are equal to two corresponding sides and the angle included between them of another triangle, then the triangles are congruent. 

ASA Congruence Criterion:

If under a correspondence, two angles and the included side of a triangle are equal to two corresponding angles and the included side of another triangle, then the triangles are congruent. 

RHS Congruence Criterion:

If under a correspondence, the hypotenuse and one side of a right-angled triangle are respectively equal to the hypotenuse and one side of another right-angled triangle, then the triangles are congruent. 

1 cm² = 100 mm²

1 m² = 10000 cm²

1 hectare = 10000 m²