Table of contents

Introduction

Basic Geometry Concepts

2.1.

Straight lines

2.2.

Angles

2.3.

Triangle

2.3.1.

Similar Triangles

2.3.2.

Congruent Triangles

2.3.3.

Some Theorems of Triangles

2.4.

Quadrilaterals

Sample Questions

Frequently Asked Questions

4.1.

List different types of geometry.

4.2.

How to remember formulas?

4.3.

What are the applications of geometry in real life?

4.4.

What does geometry deal with?

4.5.

Write the formulas for the perimeter and area of a right-angled triangle.

Conclusion

Last Updated: Mar 27, 2024

Easy

Basic Geometry Knowledge

Q: List different types of geometry.

Different types of geometry include - Euclidean Geometry, Spherical Geometry, and Hyperbolic Geometry.

Q: How to remember formulas?

An easy way to remember the formulas is by making a chart and pasting it above your study table.

Q: What are the applications of geometry in real life?

Some of the applications of geometry in real life are - Construction of buildings, Computer Graphics and Interior Design

Q: What does geometry deal with?

Geometry deals with the study of shapes, the study of sizes and the study of space.

Q: Write the formulas for the perimeter and area of a right-angled triangle.

The perimeter of a triangle is the summation of all the sides. The formula is perimeter = base + perpendicular + hypotenuse. The area of a right-angled triangle is: Area = 0.5*base*height

Author Urwashi Priya

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Introduction

Welcome Ninjas! This blog will go through a series of basic geometry knowledge, starting with the basics. We will begin the blog by introducing the classical concepts of straight lines, angles and triangles, followed by some sample questions.

Basic Geometry Concepts

Now we will start with our basic geometry knowledge:

Straight lines

Angles

A figure formed by two rays is called an angle.

If angle AOB = 90 degrees, RIGHT ANGLE.
If angle AOB < 90 degrees, ACUTE ANGLE.
If angle AOB > 90 and AOB < 180 degrees, OBTUSE ANGLE.
If angle AOB = 180 degrees, STRAIGHT ANGLE.
If two angles add to form 90 degrees, they are called complementary angles.
If two angles add to form 180 degrees, then it is called supplementary angles.

Triangle

If AB=BC=AC, the triangle is an equilateral triangle.
If AB=BC or AC=BC or AB=AC, the triangle is an isosceles triangle.
If AB!=BC!=AC, the triangle is a scalene triangle.

Area of Triangle = ½ * base * height.

Here BC is the triangle’s base, and a line from A perpendicular to BC, AD will be the triangle’s height.

Therefore

Area of the triangle = ½ * a * h

Similar Triangles

We have two concepts when comparing two triangles; one of them is the ‘Similarity of triangles’.

We call two triangles to be similar if they have the same shape, but their size may or may not be the same. This means that the angles of the triangles are congruent (equal), and the sides of the triangles are proportional.

There are several ways to determine the two triangles to be similar:

AA Similarity
If two triangles have two congruent angles, they are similar. This is known as AA (angle-angle) similarity.
SAS Similarity
When two triangles have two congruent sides, and the angle between them is congruent, they are similar. This is known as the SAS (side-angle-side) similarity.
SSS Similarity
If two triangles have three congruent sides, they are similar. This is known as SSS (side-side-side) similarity.
HL Similarity
If the hypotenuse and one of the other sides of a right-angled triangle are congruent to the hypotenuse along with the corresponding side of the other right triangle, the two triangles are similar. This is known as HL (hypotenuse-leg) similarity.

Congruent Triangles

Another concept we cover is of congruency of two triangles. We can say two triangles are congruent if one triangle can be made to superpose on the other to cover it exactly.

Below are some of the postulates for congruent triangles.

Side Angle Side (SAS) Congruence Postulate
If the two sides of one triangle and the angle between them are equal to the corresponding sides and angle of another triangle.
Angle Side Angle (ASA) Congruence Postulate
Two angles and the side between those angles of a triangle is equal to the relative two angles and the side between those two angles of another triangle.
Angle Angle Side (AAS) Congruence Postulate
Suppose any two angles and the non-included side of one triangle equal another triangle's corresponding angle and side. In that case, both triangles are said to be congruent.
Side Side Side (SSS) Congruence Postulate
All sides of a triangle are equal to the corresponding three sides of another triangle; then, both triangles are congruent.
Right Angle Hypotenuse (RHS) Congruence Postulate
If the hypotenuse and any of the sides of one right triangle are equal to the other triangle's same side and hypotenuse, then both triangles are congruent.

Some Theorems of Triangles

Some theorems which make our work easy while computing on triangles are listed below:

Carnot’s Theorem
Points D, E, and F are on the sides BC, AC and AB of a triangle named ABC. The perpendiculars corresponding to the sides of the triangle at points D, E and F are concurrent if and only if,

BD²-DC²+CE²-EA²+AF²-FB² = 0

Ceva’s Theorem
If points D, E, and F are on the sides BC, CA and AB of triangle ABC, the lines AD, BE, and CF are concurrent at point P.
Then,

BD/DC * CE/EA * AF/FB = 1

Quadrilaterals

Let us know about some properties of quadrilaterals.

Properties	Rectangle	Square	Parallelogram	Rhombus	Trapezium
All sides equal	✗	✓	✗	✓	✗
Opposite sides equal	✓	✓	✓	✓	✗
Parallel opposite sides	✓	✓	✓	✓	✓
All angles equal	✓	✓	✗	✗	✗
Equal opposite angles	✓	✓	✓	✓	✗
Sum of two adjacent angles equals a straight angle	✓	✓	✓	✓	✗
Diagonals bisect each other	✓	✓	✓	✓	✗
Diagonals bisect perpendicularly	✗	✓	✗	✓	✗

Sample Questions

Q1. The figure below shows that the triangle ABC is similar to the XYZ triangle. What is the value of cos A?

SOLUTION:

cos A = base/hypotenuse
Or AB/AC
∵ Similar triangles 
∴ AB/AC=XZ/XY
By Pythagoras' theorem, hypotenuse^2 = perpendicular^+base^2
From the figure, 
ZY2 = XY2+XZ2
XY2 = (42) - (22)
XY2 = 16 - 4
XY = √12
Cos A = XY/ZY = (√12)/4
Cos A = (2√3)/4
Cos A = √3/2

Q2. In a right-angled triangle, if one acute angle is double another. Prove that hypotenuse is double the minor side.

Given:

In △ABC, angle B is 90 degrees and ∠ACB = 2 ∠CAB.

To prove: AC=2BC

Construction: Produce CB to D such that CB=BD. Join AD.

Proof:

In △ABD and △ABC
AB=AB
∠ABD=∠ABC=90°
BD=BC
∴ By SAS, △ABD and △ABC are congruent.
∠ADB=∠ACB=2*corresponding parts of the congruent triangle
And ∠BAD=∠BAC=x
∠DAC=∠ACD=∠CDA
∴  △ADC is equilateral.
AC = DC
   = DB+BC
   = BC+BC
AC=2BC.

Q3. In the given triangle, sin x=0.8 and cos x=0.6. What is the area of the triangle?

Solution:

sin=opposite/hypotenuse
If sine x = 0.8, then 0.8 = opposite/10. 
∴ opposite = 8
cosine=adjacent/hypotenuse
If cosine x = 0.6, then 0.6 = adjacent/10. 
∴ adjacent = 6
∴ Area = ½ * base * height
       = ½ * opposite * adjacent
       = ½ * 8 * 6 = 24

Q4. In the figure below, what is the length of line segment BD? All segments on the left-hand side are 3 units, and on the right-hand side are five units.

Solution:

The question asks for the length of line segment BD. 
The five equal lengths that make up the two sides of the largest triangle tell us that we are dealing with five similar triangles. 
CA = 3+3+3+3+3 = 15 (for 5 units of side)
Similarly, CE = 25
The largest triangle has sides in the ratio of 15:25:30, and the sides of all five triangles will have an equivalent ratio. 
15/5 : 25/5 : 30/5 = 3 : 5 : 6
The reduced ratio becomes 3:5:6, which is the dimension of the smallest triangle.
To find BD, we know that the triangle has sides 6 and 10, as CB = 3*2 = 6 and CD = 5*2 =10.
This is two times as big as the smallest triangle, so the base BD = 6*2=12.

Frequently Asked Questions

List different types of geometry.

Different types of geometry include - Euclidean Geometry, Spherical Geometry, and Hyperbolic Geometry.

How to remember formulas?

An easy way to remember the formulas is by making a chart and pasting it above your study table.

What are the applications of geometry in real life?

Some of the applications of geometry in real life are - Construction of buildings, Computer Graphics and Interior Design.

What does geometry deal with?

Geometry deals with the study of shapes, the study of sizes and the study of space.

Write the formulas for the perimeter and area of a right-angled triangle.

The perimeter of a triangle is the summation of all the sides. The formula is Perimeter = base + perpendicular + hypotenuse. The area of a right-angled triangle is: Area = 0.5*base*height

Conclusion

This article taught us how to approach geometrical problems. We discussed its concepts using illustrations, diagrams, and examples.

We hope you can take away critical techniques like analysing problems by walking over the execution of the examples and finding out the pattern followed in most of the problems.