Practical Geometry

Construction of a line parallel to a given line through a point that does not lie on the line can be done by using a ruler and a pair of compasses only.

Any one of the following two properties is used to verify that the lines drawn are parallel:

When a transversal cuts two lines, such that pairs of corresponding angles are equal, then the lines have to be parallel.

When a transversal cuts two lines, such that pairs of alternate interior angles are equal, then the lines have to be parallel.

A triangle is a simple closed plane figure made up of three line segments. A triangle has three vertices, three sides and three angles.

Some of the properties of a triangle are:

Exterior angle property: The exterior angle of a triangle is equal to the sum of two opposite interior angles.

Angle sum property: Sum of all interior angles of a triangle is 180°. The sum of the lengths of any two sides of a triangle is greater than the third side.

In any right-angled triangle, the square of the length of hypotenuse is equal to the sum of the squares of the lengths of the other two sides. i.e.,Hypotenuse2 = Base2 + Altitude2

A triangle can be constructed in the following four cases:

SSS criterion: When lengths of three sides are given. SAS criterion: When lengths of two sides and the included angle are given.

ASA criterion: When two angles and the included side are given.

RHS criterion: When the length of one side and hypotenuse of a right-angled triangle are given.