NCERT Class 9 Mathematics Cheat Sheet Triangles Geometry Board Sample Problems

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Congruence

  • Two Geometric figure are said to be congruence if they are exactly same size and shape

  • Symbol used is

  • Two angles are congruent if they are equal

  • Two circle are congruent if they have equal radii

  • Two squares are congruent if the sides are equal

Triangle Congruence

Two triangles are congruent if three sides and three angles of one triangle is congruent to the corresponding sides and angles of the other

Triangle Congruence

Triangle Congruence

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Corresponding sides are equal

Corresponding angles are equal

We write this as

The above six equalities are between the corresponding parts of the two congruent triangles. In short form this is called C.P.C.T

We should keep the letters ¡n correct order on both sides

Different Criterion for Congruence of the triangles

Different Criterion for Congruence of the triangles

N

Criterion

Description

Figures and expression

1

Side angle Side (SAS) congruence

Two triangles are congruent if the two sides and included angles of one triangle is equal to the two sides and included angle

It is an axiom as it cannot be proved so it is an accepted truth

ASS and SSA type two triangles may not be congruent always

Side angle Side (SAS) congruence

Side Angle Side (SAS) Congruence

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If following condition

Then

2

Angle side angle (ASA) congruence

Two triangles are congruent if the two angles and included side of one triangle is equal to the corresponding angles and side

It ¡s a theorem and can be proved

Angle side angle (ASA) congruence

Angle Side Angle (ASA) Congruence

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If following condition

Then

3

Angle- angle side (AAS) congruence

Two triangles are congruent if the any two pair of angles and any side of one triangle is equal to the corresponding angles and side

It is a theorem and can be proved

Angle- angle side (AAS) congruence

Angle- Angle Side (AAS) Congruence

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If following condition

Then

4

Side-Side-Side (SSS) congruence

Two triangles are congruent if the three sides of one triangle is equal to the three sides of the another

Side-Side-Side (SSS) congruence

Side-Side-Side (SSS) Congruence

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If following condition

Then

5

Right angle- hypotenuse- side(RHS) congruence

Two right triangles are congruent if the hypotenuse and a side of the one triangle are equal to corresponding hypotenuse and side of the another

Right angle- hypotenuse- side(RHS) congruence

Right Angle- Hypotenuse- Side(RHS) Congruence

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If following condition

Then

Inequalities ¡n Triangles

1) In a triangle angle opposite to longer side is larger

2) In a triangle side opposite to larger angle is larger

3) The sum of any two sides of the triangle is greater than the third side

In triangle ABC

Some Important points on Triangles

Some Important points on Triangles

Orthocenter

Point of intersection of the three altitude of the triangle

Equilateral

Triangle whose all sides are equal and all angles are equal to

Median

A line Segment joining the corner of the triangle to the midpoint of the opposite side of the triangle

Altitude

A line Segment from the corner of the triangle and perpendicular to the opposite side of the triangle

Isosceles

A triangle whose two sides are equal

Centroid

Point of intersection of the three median of the triangle is called the centroid of the triangle

In center

All the angle bisector of the triangle passes through same point

Circumcenter

The perpendicular bisector of the sides of the triangles passes through same point

Scalene triangle

Triangle having no equal angles and no equal sides

Right Triangle

Right triangle has one angle equal to

Obtuse Triangle

One angle is obtuse angle while other two are acute angles

Acute Triangle

All the angles are acute