1. Introduction

Euclid, a mathematician who lived about $2300$ years ago, wrote the Elements, a book which included definitions, properties and proofs for plane geometry where two points on a plane can always be connected by a straight line. This is what we learnt at school and is a part of the curriculum for MBA entrance tests. Other forms of Geometry like Elliptic, Hyperbolic, etc. are not part of this course.

Euclidean Geometry is the part of Mathematics that deals with figures, shapes and sizes, of $0, 1$ and $2$ dimensions. One has to have a keen sense of visualisation combined with a thorough knowledge of theorems to solve Geometry problems.

In this lesson, we will start with the properties of points and lines before moving on to Triangles. Properties of polygons and circles are covered in the subsequent lessons. Properties of $3$-dimensional shapes are detailed in the Mensuration Lesson.

1.1 Revisiting Basics

Let's start with a recap of the basics of Geometry, which we learnt at school.

1.1.1 Zero-Dimensional

A point does not have any dimension (like length, breadth or height). It is the basic unit of any shape. Shape is a collection of points.
Though you might not be questioned on points directly, it is imperative for you to understand the following:
Collinear Points are points which lie on the same line.
The Vertex is a point where $2$ or more lines or curves meet.

$ $Figure 1

$ \space \space \space$Figure 2

A, B and C are the vertices in Figure 1. P, Q, R, S, T and U are the vertices in Figure 2.

Example 1

If AB = $13$ cm, BC = $8$ cm and points A, B and C are collinear, then AC = ?

(1) $13$ cm$\space \space \space$ (2) $21$ cm$\space \space \space$ (3) $13$ cm or $21$ cm$\space \space \space$ (4) $5$ cm or $21$ cm$\space \space \space$

Solution

AB is the longer side of $13$ cm length. There are two possibilities here.

Case 1: B is between A and C

AC = AB + BC $= 13 + 8$
⇒ AC = $21$ cm

Case 2: C is between A and B

AC + CB = AB
⇒ AC = AB – CB = $13 - 8$
⇒ AC = $5$ cm

Answer: (4) $5$ cm or $21$ cm

1.1.2 One-Dimensional

Definition	Figure
A Line is made up of infinite number of points and its only dimension is length. It can be infinitely extended in both directions.
A Ray is similar to a line, however, it can be infinitely extended in one direction only.
A Line Segment is a line confined between 2 fixed points. A line segment has a defined length.
Parallel Lines are lines that never meet, i.e., they are uniformly separated by the same distance throughout.
Perpendicular Lines are lines that intersect at right angles.
Concurrent Lines are lines that intersect at the same point.

1.1.3 Two-Dimensional

Definition	Figure
A Plane is a flat surface with only length and width as its dimensions. It is a figure made up of infinite number of lines.
Triangles are shapes bound by $3$ non parallel lines.
Quadrilaterals are shapes bound by $4$ lines.
A Polygon is any $n$-sided figure, where $n$ is a finite integer greater than $2$. Note: Triangles and Quadrilaterals are Polygons as well.
A Circle is a closed figure wherein the distance between the centre and any point on the surface is constant.

Intersection points of lines on a plane

Parallel lines never intersect each other on a plane. However, two non-parallel lines always intersect each other at one point on a plane.

In a plane, if we have $\bm{n}$ lines, where no $2$ lines are parallel to each other and no three lines are concurrent (i.e. passing through the same point),

Number of Intersection points = $^{n}C_{2} = \dfrac{n!}{(n - 2) ! \times 2!}$

Rationale: Any pair of these lines will intersect at one distinct point. So, number of points is the number of pairs of lines that can be selected from $\bm{n}$ lines, which is $^{n}C_{2}$.

Example 2

On a plane with $7$ lines – $a, b, c, d, e, f,$ and $g$, lines $b, c$ and $d$ are parallel while lines $e, f$ and $g$ are concurrent. At a maximum of how many points do these lines intersect in the plane?

Solution

If none of the lines are parallel or concurrent,
Number of intersection points = $^{7}C_{2} = \dfrac{7!}{5! \times 2!} = 21$

From this, we subtract intersections that did not occur due to parallel and concurrent lines.

Intersection points of 3 non-parallel and non-concurrent lines = $^{3}C_{2} = \dfrac{3!}{1! \times 2!} = 3$

$b, c$ and $d$ are parallel lines that intersect at $\bm{0}$ points instead of $\bm{3}$ points. So, total is to be reduced by $\bm{3}$.

$e, f$ and $g$ are concurrent lines that intersect at $\bm{1}$ point instead of $\bm{3}$ points. So, total is to be reduced by $\bm{2}$.

Actual Intersection Points = $21 - 3 - 2 = 16$

Answer: $16$

Maximum parts/divisions formed by n lines passing through a closed figure = $1 + \sum n$ = $1 + \dfrac{n(n + 1)}{2}$

For example, if $5$ lines are drawn in a circle, maximum number of divisions formed is $1 + \dfrac{5 \times 6}{2} = 16$

2. Angles

Where two lines or rays share a common end point, the angle is the measure of bending of one line/ray with respect to the other.

Angles are measured in degrees (denoted as $\degree$) or in radians (denoted as rad or $\degree$).

A full rotation is defined to have 360 degrees (also denoted as 360$\degree$). The length of the circular arc subtended by two lines divided by the radius of the arc is the measure of an angle in radians.

A full rotation results in a circle, whose radius is $r$ and circumference is $2\pi r$.

Angle formed by a full rotation = $\dfrac{2\pi r}{r} = 2 \pi$ rad

Note: Equating the two we get $2 \pi$ rad = $360\degree$ ⇒ $\pi$ rad $= 180\degree$

The following table contains the different types of angles and angle pairs.

Definition	Figure
Acute Angle $0\degree$ < $\theta$ < $90\degree$ $0$ rad < $\theta$ < $\pi/2$ rad
Right Angle $90\degree$ $\pi/2$ rad
Obtuse Angle $90\degree$ < $\theta$ < $180\degree$ $\pi/2$ rad < $\theta$ < $\pi$ rad
Straight Angle = $180\degree$ = $\pi$ rad
Reflex Angle $180\degree$ < $\theta$ < $360\degree$ $\pi$ rad < $\theta$ < $2\pi$ rad
Complementary angles are two angles whose sum is $90\degree$ $\angle$AEB + $\angle$CED = $90\degree$. $\therefore$ $\angle$AEB and $\angle$CED are complementary angles.
Supplementary angles are two angles whose sum is $180\degree$ $\angle$AEB + $\angle$CED = $180\degree$. $\therefore$ $\angle$AEB and $\angle$CED are supplementary angles.
Linear Pair are two adjacent angles which are on the same side of a line and equal to $180\degree$ $\angle$ADC and $\angle$BDC are on the same side of the line AB. $\therefore$ $\angle$ADC and $\angle$BDC are a linear pair.

Example 3

Two supplementary angles are in the ratio $3 : 5$. What is the difference between the two angles?

Solution

Let the two angles be $3x$ and $5x$ respectively.

Sum of supplementary angles = $180\degree$

$\therefore$ $3x + 5x$ = $180\degree$
$x = \dfrac{180}{8} = 22.5\degree$

Difference between the two angles = $5x - 3x = 2x = 2 \times 22.5 = 45\degree$

Answer: $45\degree$