## Triangles

**Triangle:** 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.

**Congruence of Triangles:** Two triangles are congruent if the sides and angles of one triangle are equal to the corresponding sides and angles of the other triangle.

If ∆PQR is congruent to ∆ABC, we write ∆PQR = ∆ABC.

Note: Congruent triangles corresponding parts are equal and we write in short ‘CPCT’ for Corresponding Parts of Congruent Triangles.

Criteria for Congruence of Triangles.

**SAS congruence rule:** Two triangles are congruent if two sides and the included angle of one triangle are equal to the sides and the included angle of the other triangle.
**ASA congruence rule:** Two triangles are congruent if two angles and the included sides of one triangle are equal to two angles and the included side of another triangle.
**AAS congruence rule:** Two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal.
**SSS congruence rule:** Two triangles are congruent if three sides of one triangle are equal to the sides of the other triangle.
**RHS congruence rule:** If in two right triangles, hypotenuse and one side of a triangle are equal to the hypotenuse and one side of other triangles, then the two triangles are congruent.

Properties of a Triangle

**Isosceles triangle:** A triangle in which two sides are equal is called an isosceles triangle. So, ∆ABC is an isosceles triangle with AB = AC.

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

i.e., ∠B = ∠C
- The sides opposite to equal angles of a triangle are equal. i.e., AB = AC

Inequalities in a Triangle

- If two sides of a triangle are unequal, the angle opposite to the longer side is larger (or greater).
- In any triangle, the side opposite to the larger (or greater) angle is longer (converse of (i)).
- The sum of any two sides of a triangle is greater than the third side, i.e., AB + BC > CA.

NCERT Solutions Ch- 7 TextBook

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