** Sunil Jakhar**

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## Plane Geometry

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### Triangle

A triangle is a geometrical figure that is contained by three straight lines. Every triangle has three sides and three angles. It is most basic as well as the most important shape. The triangle serves as a basis to understand all other geometrical figures and their properties.

The triangle is the only rigid figure among all polygons as there is no scope for “wiggling”. Unlike all other higher polygons, a triangle can be completely defined using the least number of parameters.

Triangles are generally classified under four categories:

**Equilateral**: The triangles which have all sides of equal in length.**Isosceles:**The triangles which have two of the sides of equal in length.**Scalene**: The triangles which have all sides of different lengths.**Right Angled:**The triangles which have one of the angles as the right angle. The side opposite to the right angle is referred to as the hypotenuse.

### Congruence of Triangles

Geometrical objects are said to be congruent when they have the same shape and size. ‘Congruent’ can be understood as the geometrical equivalent of the common term ‘equal’. The necessary and sufficient conditions to decide congruence in case of triangles are as follows:

**SAS**(Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.**SSS**(Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.**ASA**(Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.**AAS**(Angle-Angle-Side): If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent.**RHS**(Right-angle-Hypotenuse-Side): If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent.

### Similarity of Triangles

Geometrical objects are said to be similar when they have the same shape but not necessarily the same size. All geometric figures that can be seen as merely scaled up or scaled down versions of each other can be called similar. All circles are similar irrespective of their size because they all have the same shape.

In case of triangles, the conditions to determine similarity are as follows:

- If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar.
- If corresponding sides of triangles are in proportion, then the triangles are similar.