NCERT Class 9 Mathematics Cheat Sheet Triangles Geometry Board Sample Problems (For CBSE, ICSE, IAS, NET, NRA 2022)

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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

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 in correct order on both sides

Different Criterion for Congruence of the triangles

Different Criterion for Congruence of the Triangles
NCriterionDescriptionFigures and expression
1Side angle Side (SAS) congruenceTwo 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
If following condition

Then

2Angle side angle (ASA) congruenceTwo triangles are congruent if the two angles and included side of one triangle is equal to the corresponding angles and side

It is a theorem and can be proved

Angle Side Angle (ASA) Congruence
If following condition

Then

3Angle- angle side (AAS) congruenceTwo 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
If following condition

Then

4Side-Side-Side (SSS) congruenceTwo triangles are congruent if the three sides of one triangle is equal to the three sides of the another
Side-Side-Side (SSS) Congruence
If following condition

Then

5Right angle- hypotenuse- side (RHS) congruenceTwo 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
If following condition

Then

Inequalities in 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
OrthocenterPoint of intersection of the three altitude of the triangle
EquilateralTriangle whose all sides are equal and all angles are equal to
MedianA line Segment joining the corner of the triangle to the midpoint of the opposite side of the triangle
AltitudeA line Segment from the corner of the triangle and perpendicular to the opposite side of the triangle
IsoscelesA triangle whose two sides are equal
CentroidPoint of intersection of the three median of the triangle is called the centroid of the triangle
In centerAll the angle bisector of the triangle passes through same point
CircumcenterThe perpendicular bisector of the sides of the triangles passes through same point
Scalene triangleTriangle having no equal angles and no equal sides
Right TriangleRight triangle has one angle equal to
Obtuse TriangleOne angle is obtuse angle while other two are acute angles
Acute TriangleAll the angles are acute