# Aptitude - Geometry

## Point

A point is an exact location

## Line Segment

The straight path between two points A and B is called a line segment AB. A line segment has two end points.

## Ray

On extending a line segment AB indefinitely in one direction we get the ray AB. Ray AB has one end point, namely A.

## LINE

A line segment AB extended indefinitely in both directions is called line AB.

A line contains infinitely many points.

Through a given points , infinitaly many lines can be drawn.

One and only one line can be drawn to pass through two given points A and B.

Two line meet in a point.

Two planes meet in a line.

### Collinear

In the given figure, the points A,B,C are collinear.

### Concurrent Lines

Three or more lines intersecting at the same points are called concurrent lines.

## Angle

Two rays OA and OB having a common end points O form angle AOB, written as ∠AOB

### Measure of an Angle

The amount of turning from OA to OB is called the measure of ∠AOB written as m(∠AOB).

### An angle of 360°

If a ray OA starting from its original position OA , rotates about O in anticlockwise direction and after a complete rotation comes back to its original position , then we say that it has rotated through 360. This complete rotation is divided into 360° equal parts. Then, each part is called 1 degree , written as 1°

1° = 60 minutes, written as 60'

1 minute = 60 seconds, written as 60"

### Types of Angle

**Right angle**- An angle whose measure is 90° is called a right angle.**Acute angle**- An angle whose measure is less than 90° is called an acute angle.**Obtuse angle**- An angle whose measure is more than 90° but less than 180°, is called an obtues angle.**Straight angle**- An angle whose measure is 180° is called a Straight angle.**Reflex angle**- An angle whose measure is more than 180° but less than 360°, is called a Reflex angle.**Complete angle**- An angle whose measure is 360°, is called a complete angle.**Equal angle**- Two angles are said to be equal , if they have the same measure.**Complementary angle**Two angles are said to be complementary if the sum of their measures is 90. For example, angles measuring 65° and 25° are complementary angle.**Supplementary angle**- Two angle are said to be supplementary if the sum of their measures is 180°. For example, angles measures 70° and 110° are supplementary.**Adjacent angle**- Two angles are called adjacent angle if they have the same vertex and a common arm such that non-common arms are on either side of the comman arm. In the given figure , ∠AOC and ∠BOC are adjacent angle.

## Important Results

If a ray stands on a line , than the sum of two adjacent angle so formed is 180° In the given figure , ray CP stands on line AB.

∴ ∠ACD + ∠BCD = 180°.

The sum of all angle formed on the same side of a line at a given point on the line is 180°. In the given figure four angle are formed on the same side of AOB.

∴ ∠AOE + ∠EOD + ∠DOC + ∠COD = 180°.

The sum of all angle around a point is 360° In the given figure five angle are formed around a point O.

∴∠AOB + ∠BOC + ∠COD + ∠DOE + ∠EOA=360°.

## Vertically Opposite Angles

If two lines A Band CD intersect at a point O, then AOC , BOD and BOC , AOD are two pair of vertically opposites angle Vertically opposite angle are always equal.

∴ ∠AOC = ∠BOD and ∠AOD = ∠BOC

## Parallel Lines

If two lines lie in the same plane and do not intersect when produced on either side then such lines are said to be paralleled and we write , L||m.

## Traversal line cutting parallel lines

Let two parallel lines AB and CD be cut by a transversal EF. Then

Corresponding angle are equal

(∠1 = ∠5), (∠4= ∠8 ), (∠2 = ∠6) , (∠3 = ∠7)

Alternate interior angles are equal.

(∠3 =∠5 ) and (∠4 =∠6 )

Consective interior angles are supplementary

∠4+∠5 = 180° and ∠3 +∠6 = 180°.

## Triangle

A figure bounded by three straight lines is called a triangle. In the given figure , we have ∆ABC; ∆ABC having three vertices A,B,C. In has three angles, namely ∠A,∠B and ∠C. It has three sides , namely AB, AC and BC.

## Types of Triangle

A triangle having all sides equal is called an equilateral triangle.

A triangle having two sides equal, is called an isosceles triangle.

A triangle having all sides of different lengths,is called a scalene triangle.

A triangle one of whose angles measures 90°,is called a right triangle.

A triangle one of whose angle lies between 90° and 180° is called an obtuse triangle.

A triangle each of whose angle is acute, is called an acute triangle.

The sum of all sides of a triangle is called the perimeter of the triangle.

The sum of two sides of a triangle is greater than the third side.

In a right angled ABC in which ∠B = 90°, we have AC

^{2}=AB^{2}+BC^{2}. This is called Pythagoras Theorem.

## Quadrilateral

A figure bounded by four straight line is called a quadrilateral. The sum of all angles of a quadrilateral is 360°.

**Rectangle**- A quadrilateral is called a rectangle, if its opposite side are equal and each of its angle is 90°. In given fig. ABCD is a rectangle.**Square**- A quadrilateral is called a square, if all of its sides are equal and each of its angles measures 90°. In given fig. ABCD is square in which AB = BC = CD = DA.**Parallelogram**- A quadrilateral is called a parallelogram, if its opposite sides are parallel. In given fig. ABCD is a parallelogram in which AB = DC & AD = BC.**Rhombus**- A parallelogram having all sides equal is called a rhombus. In given fig. ABCD is a rhombus in which AB =BC =CD=DA, AB || DC and AD || BC.

## Important Facts

A quadrilateral is a rectangle if opposite sides are equal and its diagonals are equal.

A quadrilateral is a Square if all sides are equal and the diagonal are equal.

A quadrilateral is a parallelogram, if opposite sides are equal.

A quadrilateral is a parallelogram but not a rectangle, if opposite sides are equal but the diagonals are not equal.

A quadrilateral is a rhombus but not a square if all their sides are equal and the diagonals are not equal.

## Results on Quadrilateral

In a parallelogram, we have

Opposite sides are equal.

Opposite angles are equal.

Each diagonal bisects the parallelogram.

Diagonals of a parallelogram bisect each other.

Diagonals of a rectangle are equal.

Diagonals of a rhombus bisect each other at right angles.

## Results on Circle

The perpendicular from the center to a chord bisects the chord.

There is one and only one circle passing through three non collinear points.

Angle in a semi circle is a right angle.

Opposite angles of a cyclic quadrilateral are supplementary.

Angle in the same segment of a circle is equal.

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Two tangent to a circle from a point outside it are equal.

If PT is a tangent to a circle and PAB is a secant, Then PA x PB= PT

^{2}