# NCERT Class 9 Mathematics Cheat Sheet Triangles Geometry Board Sample Problems (For CBSE, ICSE, IAS, NET, NRA 2023)

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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

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

 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 angleIt is an axiom as it cannot be proved so it is an accepted truthASS and SSA type two triangles may not be congruent always Side Angle Side (SAS) CongruenceIf following conditionThen 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 sideIt is a theorem and can be proved Angle Side Angle (ASA) CongruenceIf following conditionThen 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 sideIt is a theorem and can be proved Angle- Angle Side (AAS) CongruenceIf following conditionThen 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) CongruenceIf following conditionThen 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) CongruenceIf following conditionThen

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

 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