Notes of Triangles

Congruence of Triangles

In a pair of triangles if all three corresponding sides and three corresponding angles are exactly equal, then the triangles are said to be congruent.

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Congruent triangles

In congruent triangles, the corresponding parts are equal and are written as CPCT (Corresponding part of congruent triangle).

Two figures are congruent, if they are of the same shape and of the same size. 

Two circles of the same radii are congruent. 

Two squares of the same sides are congruent. 

If two triangles ABC and DEFare congruent under the corresponding A ↔ D, BE and CF, then symbolically, it is expressed as ∆ ABC ∆ DEF. 

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In the figure, ∆ ABC ≅ ∆ DEF

SSS Criteria for Congruency

  • If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.
  • If all sides are exactly same, then their corresponding angles must also be exactly same.

SAS Criteria for Congruency

Axiom : Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding  sides and the included angle of the other triangle.

ASA Criteria for Congruency

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

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Included side is 4cm

AAS Criteria for Congruency

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

Why SSA and AAA congruency rules are not valid?

  • SSA or ASS test is not a valid test for congruency as the angle is not included between the pairs of equal sides.-
  • The AAA test also is not a valid test as even though 2 triangles can have all three same angles, the sides can be of differing lengths. This becomes a test for similarity (AA).
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Angles of a triangle

RHS Criteria for Congruency

  • 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.
  • RHS stands for Right angle – Hypotenuse – Side.

Properties of Isosceles triangle

If 2 sides of the triangle are equal, the angles opposite those sides are also equal and vice versa.

Criteria for Congruency of triangles

The criteria for congruency of triangles are : 

In congruent triangles, corresponding parts are equal. We write in short ‘CPCT’ for corresponding parts of congruent triangles. 

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. 

SAS congruence rule holds but not ASS or SSA rule. 

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. 

AAS congruence rule 

Two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal.

SSS congruent rule  

If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.

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. 

  1. SAS
  2. ASA
  3. AAS
  4. SSS
  5. RHS

symbolically, it is expressed as ΔABC ≅ ΔXYZ

Inequalities in Triangles

Relationship between unequal sides of triangle and the angles opposite to it.

If 2 sides of a triangle are unequal, then the angle opposite to the longer side will be larger than the angle opposite to the shorter side.

Triangle inequality

The sum of the lengths of any two sides of a triangle must be greater than the third side.

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

Angles opposite to equal sides of an isosceles triangle are equal. 

The sides opposite to equal angles of a triangle are equal. 

If two sides of a triangle are unequal, the angle opposite to the longer side is greater

In any triangle, the side opposite to greater angle is longer

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

The difference between any two sides of a triangle is less than the third side.