# Geometry

Geometry is the branch of mathematics that deals with the properties, measurement and relations of points, lines, angles, surfaces and solids.

Geometry is such a vast topic and therefore it is almost an impossible endeavour to gather or compile all the concepts pertaining to this in a single book. Therefore, we will be discussing only the relevant concepts with respect to the type of questions that are posed in the CSIR-NET examination. With a proper understanding of these concepts, you will be able to attempt any geometrical questions in the CSIR-NET examination.

Geometry can be divided into plane geometry and solid geometry.

### Plane Geometry

A plane is any two-dimensional surface such as circles and squares. Plane geometry is all about figures in a plane.

Let us discuss some important aspects in plane geometry.

1. Line

Types of lines

• Intersecting lines

Two or more lines that meet at a point are called intersecting lines. The common point is called the point of intersection.

• Concurrent Lines

Three or more lines in a plane are concurrent if they intersect at a single point.

• Perpendicular Lines

A line is said to be perpendicular to another line if the two lines intersect at a right angle (90O angle).

Parallel Lines
intersect or touch at any point are called parallel lines.

Parallel lines and the transversal

A transversal is a line that intersects two or more other lines. When it intersects parallel lines, the following angles are congruent (equal).

Corresponding angles – Angles that are in the same position on each line.

Alternate interior angles – Angles on the opposite sides of the transversal and on the interior of the parallel lines.

Alternate exterior angles – Angles on opposite sides of the transversal and on the exterior of the parallel lines.

Vertically opposite angles – The pairs of opposite angles formed between each line and the transversal.

Note: Angles on one side of a straight line will always add to 180 degrees [Linear Pair].

1. Triangle

A triangle is a closed figure with three sides and three vertices.

Types of triangles

• Right triangles

A triangle with a right angle (That is, 90o) is called a right triangle.

Pythagoras Theorem

In a right triangle,

(Hypotenuse)2 = (Base)2 + (Height)2

The numbers that satisfy this relation are called Pythagorean triplets.

Example: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61),

(12, 35, 37), (15, 20, 25) etc.

Note: If you see a triangle with sides exactly as any of these Pythagorean triplets, you can be sure that the triangle is a right triangle.

• Isosceles triangles

Triangles with two equal sides are called isosceles triangles.

The angles opposite to the equal sides are also equal.

• Equilateral Triangles

Triangles with three equal sides are called equilateral triangles.

For equilateral triangles, each of the three angles are also equal to 60o.

• Sum of the interior angles of a triangle is 180o (Angle Sum Property)
• The sum of any two sides of a triangle is greater than the third side.
• The diagonal of a square divides the square into two isosceles right triangles.
• Median – A median of a triangle is a line segment from one vertex to the mid-point on the opposite side of the triangle.

Centroid – A centroid of a triangle is the point where the three medians of the triangle meet.

• Orthocenter – In a triangle, the orthocenter is the point of intersection of the altitudes.
• Incircle – The incircle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is called the triangle’s incenter.
• Circumcircleof a triangle is a circle, which passes through all the vertices of the triangle. The center of this circle is called the circumcenter and its radius is called the circumradius.
• Circle

Circle is the set of all points in a plane that are at a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius. The diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle.

• Chord

Chord is the line that joins two points on a circle. Diameter is the largest chord.

Arc

An arc of a circle is two points on the circle and the continuous part of the circle between the two points.

If the arc measures xo, the arc length is (2πr).

• Tangent

A tangent to a circle is a line that touches the circle at exactly one point, never entering the circle’s interior.

Important circle theorems

1. A tangent and radius meet at right angles.
2. Two tangents from a common exterior point to a circle are equal in length.
3. The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at the circumference of the circle.
4. Angles subtended by an arc in the same segment of a circle are equal.
5. The angle subtended in a semicircle is a right angle (90o).
• Parallelogram

A parallelogram is a quadrilateral (closed figure with four sides) with two pairs of parallel sides.

Important properties of Parallelogram

• Opposite sides are parallel and equal
• Opposite angles are equal.
• The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
• Diagonals of a parallelogram bisect each other. That is, they are divided in their intersection point into two.
• Parallelogram law – The sum of squares of diagonals is equal to the sum of squares of four sides.
1. Rhombus

A parallelogram with all sides equal is called a rhombus.

Important properties of Rhombus

• Opposite sides are parallel and all sides are equal.
• Opposite angles are equal.
• The diagonals of a rhombus are perpendicular to each other.
• Diagonals of a rhombus bisect each other.
• As Rhombus is a special parallelogram, it also follows the parallelogram law.

“For detailed theory, refer the book “CSIR-NET General Aptitude – A New Outlook”

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