Chapter 7.Triangles

Definition:- a closed figure formed by three intersecting lines is called a triangle. (‘Tri’ means ‘three’). A triangle has three sides, three angles and three vertices. For example, in triangle ABC, denoted as ABC ; AB, BC, CA are the three sides, <A, <B, <C are the three angles and A, B, C are three vertices

2 Congruence of Triangles

Congruent figures (‘congruent’ means equal in all respects or figures whose shapes and sizes are both the same).

Now, draw two circles of the same radius and place one on the other. They cover each other completely and we call them as congruent circles.

Note that in congruent triangles corresponding parts are equal and we write in short ‘CPCT’ for corresponding parts of congruent triangles.

3 Criteria for Congruence of 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.

So, SAS congruence rule holds but not ASS or SSA rule.

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

So, two triangles are congruent if any two pairs of angles and one pair of

corresponding sides are equal. We may call it as the AAS Congruence Rule.

Properties of Triangles

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.

5 Criteria for Congruence of Triangles

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.

Note that RHS stands for Right angle – Hypotenuse – Side.

6 Inequalities in a Triangle

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.

Summary

- 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 PQR are congruent under the correspondence A P,

B Q and C R, then symbolically, it is expressed as ABC PQR. - If two sides and the included angle of one triangle are equal to two sides and the included

angle of the other triangle, then the two triangles are congruent (SAS Congruence Rule). - If two angles and the included side of one triangle are equal to two angles and the

included side of the other triangle, then the two triangles are congruent (ASA Congruence

Rule). - If two angles and one side of one triangle are equal to two angles and the corresponding

side of the other triangle, then the two triangles are congruent (AAS Congruence Rule). - Angles opposite to equal sides of a triangle are equal.
- Sides opposite to equal angles of a triangle are equal.
- Each angle of an equilateral triangle is of 60°.
- If three sides of one triangle are equal to three sides of the other triangle, then the two

triangles are congruent (SSS Congruence Rule). - If in two right triangles, hypotenuse and one side of a triangle are equal to the hypotenuse

and one side of other triangle, then the two triangles are congruent (RHS Congruence

Rule). - In a triangle, angle opposite to the longer side is larger (greater).
- In a triangle, side opposite to the larger (greater) angle is longer.
- Sum of any two sides of a triangle is greater than the third side.