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Lines & Triangles
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CAT 2025 Lesson : Lines & Triangles - Lines & Angles

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1. Introduction

Euclid, a mathematician who lived about
23002300 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
0,10, 1 and 22 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
33-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
22 or more lines or curves meet.


Figure 1

   \space \space \spaceFigure 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 = 1313 cm, BC = 88 cm and points A, B and C are collinear, then AC = ?

(1)
1313 cm   \space \space \space (2) 2121 cm   \space \space \space (3) 1313 cm or 2121 cm   \space \space \space (4) 55 cm or 2121 cm   \space \space \space

Solution

AB is the longer side of 1313 cm length. There are two possibilities here.

Case 1: B is between A and C

AC = AB + BC =13+8= 13 + 8
⇒ AC =
2121 cm


Case 2: C is between A and B

AC + CB = AB
⇒ AC = AB – CB = 13813 - 8
⇒ AC =
55 cm



Answer: (4)
55 cm or 2121 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 33 non parallel lines.




Quadrilaterals are shapes bound by 44 lines.




A Polygon is any nn-sided figure, where nn is a finite integer greater than 22.

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
n\bm{n} lines, where no 22 lines are parallel to each other and no three lines are concurrent (i.e. passing through the same point),

Number of Intersection points =
nC2=n!(n2)!×2!^{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
n\bm{n} lines, which is nC2^{n}C_{2}.

Example 2

On a plane with 77 lines – a,b,c,d,e,f,a, b, c, d, e, f, and gg, lines b,cb, c and dd are parallel while lines e,fe, f and gg 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 = 7C2=7!5!×2!=21^{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 =
3C2=3!1!×2!=3^{3}C_{2} = \dfrac{3!}{1! \times 2!} = 3

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

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

Actual Intersection Points =
2132=1621 - 3 - 2 = 16

Answer:
1616


Maximum parts/divisions formed by n lines passing through a closed figure =
1+n1 + \sum n = 1+n(n+1)21 + \dfrac{n(n + 1)}{2}

For example, if
55 lines are drawn in a circle, maximum number of divisions formed is 1+5×62=161 + \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
rr and circumference is 2πr2\pi r.

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

Note: Equating the two we get
2π2 \pi rad = 360°360\degreeπ\pi rad =180°= 180\degree

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

Definition Figure
Acute Angle
0°0\degree < θ\theta < 90°90\degree
00 rad < θ\theta < π/2\pi/2 rad


Right Angle
90°90\degree
π/2\pi/2 rad


Obtuse Angle 90°90\degree < θ\theta < 180°180\degree
π/2\pi/2 rad < θ\theta < π\pi rad


Straight Angle = 180°180\degree = π\pi rad


Reflex Angle 180°180\degree < θ\theta < 360°360\degree
π\pi rad < θ\theta < 2π2\pi rad


Complementary angles are two angles whose sum is 90°90\degree \angleAEB + \angleCED = 90°90\degree. \therefore \angleAEB and \angleCED are complementary angles.


Supplementary angles are two angles whose sum is 180°180\degree

\angleAEB + \angleCED = 180°180\degree.
\therefore \angleAEB and \angleCED are supplementary angles.


Linear Pair are two adjacent angles which are on the same side of a line and equal to 180°180\degree

\angleADC and \angleBDC are on the same side of the line AB.
\therefore \angleADC and \angleBDC are a linear pair.


Example 3

Two supplementary angles are in the ratio 3:53 : 5. What is the difference between the two angles?

Solution

Let the two angles be 3x3x and 5x5x respectively.

Sum of supplementary angles =
180°180\degree

\therefore 3x+5x3x + 5x = 180°180\degree
x=1808=22.5°x = \dfrac{180}{8} = 22.5\degree

Difference between the two angles =
5x3x=2x=2×22.5=45°5x - 3x = 2x = 2 \times 22.5 = 45\degree

Answer:
45°45\degree


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