Page 2 :
Revision Notes on Basic Geometrical Ideas, Introduction to Basic Geometrical Ideas, The term ‘Geometry’ is derived from the Greek word ‘Geometron’. This has 2, equivalents. ‘Geo’ means Earth and ‘metron’ means Measurement., , Points, It is a position or location on a plane surface, which are denoted by a single, capital letter., , Line Segment, It is a part of a line with the finite length and 2 endpoints., The points A and B are called the endpoints of the segment., It is named as:, , Line, It is made up of infinitely many points with in infinite length and no endpoint., , It extends indefinitely in both directions., Named as:, Or sometimes, , Intersecting Lines, The two lines that share one common point are called Intersecting Lines., 1/28

Page 3 :
This shared point is called the point of intersection., , Here, line l and m are intersecting at point C., Real life example of intersecting lines:, , Parallel Lines, Two or more lines that never intersect (Never cross each other) are called Parallel, Lines., , Real life examples of parallel lines:, , Ray, It is a part of a line with one starting point whereas extends endlessly in one, direction., Real life examples of the ray are:, , 2/28

Page 4 :
Curves, Anything which is not straight is called a curve., (, 1., , Simple Curve – A curve that does not cross itself., , 2., , Open Curve – Curve in which its endpoints do not meet., , 3., , Closed Curve – Curve that does not have an endpoint and is an enclosed figure., , A closed curve has 3 parts which are as follows, , 1., , Interior of the curve, , It refers to the inside/inner area of the curve., The blue coloured area is the interior of the figure., 3/28

Page 5 :
2., , The exterior of the curve., , It refers to the outside / outer area of the curve., The point marked A depicts the exterior of the curve., 3., , The boundary of the curve, , It refers to the dividing line thus it divides the interior and exterior of the curve., The black line which is dividing the interior and exterior of the curve is the, boundary., The interior and boundary of the curve together are called the curves “region”., , Polygons, It is a 2d closed shape made of line segments / straight lines only., , 4/28

Page 6 :
Sides –It refers to the line segments which form the polygon, as in the above, figure AB, BC, CD, DA are its sides., Vertex – Point where 2 line segments meet, as in the above figure A, B, C and D are, its vertices., Adjacent Sides – If any 2 sides share a common endpoint they are said to be, adjacent to each other thus called adjacent, sides, as in the above gure AB and BC, BC and CD, CD and DA, DA, and AB are adjacent sides., Adjacent Vertices – It refers to the endpoints of the same side of the polygon. As in, the above figure A and B, B and C, C and D, D and A are adjacent vertices., Diagonals – It refers to the joins of the vertices which are not adjacent to each, other. As in the above figure, AC and BD are diagonals of the polygon., , Angles, A gure formed from 2 rays which share a common endpoint is called Angle., , The rays forming the angle are known as its arms or sides., The common endpoint is known as its vertex., An angle is also associated with 3 parts, 1., , Interior - It refers to the inside/inner area., , The green coloured area is the interior of the angle., 2., , Angle/boundary - It refers to the arms of the angle., , The red point is on the arm of the angle., 3., , Exterior - It refers to the outside / outer area., , The blue point depicts the exterior of the gure., 5/28

Page 7 :
Naming an Angle, While naming an angle the letter depicting the vertex appears in the middle., Example, , ., Example, +, ), , Triangle, It is a 3 sided polygon. It is also the polygon with the least number of the sides., , Vertices: A, B and C, Sides: AB, BC and CA, Angles: A, B and C, , 6/28

Page 8 :
Here, the light blue coloured area is the interior of the angle., The black line is the boundary., Whereas, the dark blue area is the exterior of the angle., , Quadrilaterals, It is a 4 sided polygon, , Vertices: A, B, C, D, Sides: AB, BC, CD, D A, Angle: ∠A, ∠B, ∠C, ∠D, Opposite Sides: AB and DC, BC and AD, Opposite Angles: B and D, A and C, Adjacent Angles∠A and ∠B, ∠B and ∠C, ∠C and ∠D, ∠D and ∠A., , Circles, It is a simple closed curve and is not considered aBlo) a polygon., , Parts of Circles, , 1., , 2., , Radius – It is a straight line connecting the centre of the circle to the boundary of, the same. Radii is the plural of ‘radius’., Diameter –It is a straight line from one side of the circle to the other side passing, through the centre., 7/28

Page 9 :
3., 4., , 5., , 6., , 7., , 8., , 9., , Circumference – It refers to the boundary of the circle., Chord - Any line that connects two points on the boundary of the circle is called, Chord. Diameter is the longest chord., , Arc – It is the portion of the boundary of the circle., , Interior of the Circle – Area inside the boundary of the circle is called the Interior of, the Circle., The Exterior of the Circle – Area outside the boundary of the circle is called the, Exterior of the Circle., , Sector- It is the region in the interior of a circle enclosed by an arc on one side and, a pair of radii on the other two sides., Segment - It is the region in the interior of the circle enclosed by an arc and a, chord., , 8/28

Page 10 :
), , Semi-circle, A diameter divides the circle into two semi-circles. Hence the semicircle is the half of the, circle, which has the diameter as the part of the boundary of the semicircle.

Page 11 :
10/28

Page 13 :
An angle is formed when two rays are joined together at a common point. The common point here is called node or, vertex and the two rays are called arms of the angle. The angle is represented by the symbol ‘∠’. The word angle came, from the Latin word “Angulus”. Learn more about lines and angles here., The angle is usually measured in degrees, using a protractor. Degrees 30°, 45°, 60°, 90°, 180° shows different angles, here. The types of angles are based on the values of angles in degrees., We can also represent angles in radians, i.e., in terms of pi (π). 180 degrees is equal to π in radians., , Definition, An angle is a form of geometrical shape, that is constructed by joining two rays to each other at their end-points. The, angle can also be represented by three letters of the shape that define the angle, with the middle letter being where the, angle actually is (i.e.its vertex). Angles are generally represented by Greek letters such as θ, α, β, etc., , Eg. ∠ABC, where B is the given angle., Angle measurement terms are – degree °, radians or gradians., The amount of rotation about the point of intersection of two planes (or lines) which is required to bring one in, correspondence with the other is called an Angle., , 14,180, , Types of Angles, There are majorly six types of angles in Geometry. The names of all angles with their properties are:, , 12/28

Page 14 :
•, , Acute Angle: It lies between 0° to 90., , •, , Obtuse Angle: It lies between 90° to 180°, , •, , Right Angle: The angle which is exactly equal to 90°, , •, , Straight Angle: The angle which is exactly equal to 180°, , •, , Reflex Angle: The angle which is greater than 180 degrees and less than 360 degrees, , •, , Full Rotation: The complete rotation of angle equal to 360 degrees, , Note: Sometimes full rotation is not considered as a kind of angle. Therefore, in such cases, we consider there are five, types of angles., , Type of angles, , Description, , Acute Angle, , < 90°, , Obtuse Angle, , > 90°, , Right Angle, , = 90°, , Straight Angle, , =180°, , Reflex Angle, , >180°, , Full rotation/complete angle, , =360°, , Interior and Exterior Angles, In case of a polygon, such as a triangle, quadrilateral, pentagon, hexagon, etc., we have both interior and exterior, angles., •, , Interior angles are those that lie inside the polygon or a closed shape having sides and angles., , •, , Exterior angles are formed outside the shape, between any side and line extended from adjacent sides., , For example, an image of a pentagon is given here, representing its interior angles and exterior angles.

Page 15 :
Positive & Negative Angles, •, , Positive Angle- An Angle measured in Anti-Clockwise direction is Positive Angle., , •, , Negative Angle- An angle measured in Clockwise direction is Negative Angle., , 14/28

Page 16 :
Parts of Angles, , •, , Vertex- The corner points of an angle is known as Vertex. It is the point where two rays meet., , •, , Arms– The two sides of angle, joined at a common endpoint., , •, , Initial Side – It is also known as the reference line. All the measurements are done taking this line as the, reference., , •, , Terminal Side- It is the side (or ray) up to which the angle measurement is done., , Angle Measurement, To measure everything in this world, we need a unit in a similar angle measurement requires three units of, measurement :, , Degree of an Angle, It is represented by ° (read as a degree). It most likely comes from Babylonians, who used a base 60 (Sexagesimal), number system. In their calendar, there was a total of 360 days. Hence, they adopted a full angle to be 360°. First, they, tried to divide a full angle into angles using the angle of an equilateral triangle. Later, following their number system, (base 60), they divided 60° by 60 and defined that as 1°. Sometimes, it is also referred to as arc degree or arc-degree, which means the degree of an arc., An angle is said to be equal to 1° if the rotation from the initial to the terminal side is equal to 1/360 of the full rotation., A degree is further divided into minutes and seconds. 1′ (1 minute) is defined as one-sixtieth of a degree and 1” (1, second) is defined as one-sixtieth of a minute. Thus,, 1°= 60′ = 3600”

Page 17 :
Angle Measurement in Degrees, , What is an angle?, An angle is a geometrical figure formed by two rays, when joint at a single point. The two rays are known as, arms or sides of angle and the common point is the vertex., , What are the six types of angles?, The six major types of angles are:, Acute angle, Obtuse angle, Right angle, Straight angle, Reflex angle, Full rotation, , How angles are measured?, Angles are usually measured in degrees. We can use a measuring instrument, i.e. protractor, to measure any, unknown angle., , What is a zero angle?, An angle with zero degree measurement is called zero angle., , Can a triangle have two 90 degree angles?, A triangle cannot have two 90 degrees or right angles, because by the angle sum property of the triangle, we, know that, sum of all the three angles of a triangle is equal to 180 degrees. If two angles are of 90 degrees,, then the third angle has to be zero, which is not possible., , 16/28

Page 18 :
Supplementary Angles, , Supplementary angles refer to the pair of angles that always sum up to 180°. These two, angles are called supplements of each other. The word "supplementary" comes from two, Latin words "Supplere" and "Plere" where "Supplere" means "supply”, whereas "Plere", means "fill". So, "supplementary" means "something when supplied to complete a thing"., In this lesson, we will explore the world of supplementary angles, which helps in solving, various geometry problems. The journey will take us through supplementary angles, and the, difference between complementary angles and supplementary angles., What are Supplementary Angles?, , Two angles are said to be supplementary angles if they add up to 180 degrees., Supplementary angles form a straight angle (180 degrees) when they are put together. In, other words, angle 1 and angle 2 are supplementary, if Angle 1 + Angle 2 = 180°. In this, case, Angle 1 and Angle 2 are called "supplements" of each other., Observe the following figure in which 130° + 50° = 180°. Hence, by the definition of, supplementary angles, these two angles are supplementary.

Page 19 :
Adjacent and Non-Adjacent Supplementary Angles, , Supplementary angles can either be adjacent or non-adjacent. So, there are two types of, supplementary angles. Each of these types of supplementary angles is explained below., •, , Adjacent supplementary angles, , •, , Non-adjacent supplementary angles, , Adjacent Supplementary Angles, Two supplementary angles with a common vertex and a common arm are said to be, adjacent supplementary angles., Example:, , Here, ∠COB and ∠AOB are adjacent angles as they have a common vertex, O, and a, common arm OB. They also add up to 180 degrees, that is, ∠COB + ∠ AOB = 70° + 110° =, 180°. Hence, these two angles are adjacent supplementary angles., , 18/28

Page 20 :
Non-adjacent Supplementary Angles, Two supplementary angles that are NOT adjacent are said to be non-adjacent, supplementary angles., Example:, , Here, ∠ABC and ∠PQR are non-adjacent angles as they neither have a common vertex nor, a common arm. They also add up to 180 degrees, that is, ∠ABC+ ∠PQR = 79° + 101° = 180°., Hence, these two angles are non-adjacent supplementary angles. Non-adjacent, supplementary angles, when put together, form a straight angle., Complementary and Supplementary Angles, , Supplementary and complementary angles are those angles that exist in pairs. While, supplementary angles add up to 180°, complementary angles add up to 90°. These angles, have numerous real-time applications, the most common being the crossroads. The, following table shows the difference between supplementary and complementary angles.

Page 21 :
The supplementary vs complementary angles table., Complementary Angles, , Supplementary Angles, , Two angles are said to be, , Two angles are said to be, supplementary if they add up, to 180 degrees., , complementary if they add, up to 90 degrees., Complement of an angle x is, , Supplement of an angle x is, (180 - x), °, , (90 - x)°, , Tips on Supplementary angles vs Complementary angles, Here is a short trick for you to understand the difference between supplementary angles and, complementary angles., •, , "S" is for "Supplementary" and "S" is for "Straight." Hence, you can, remember that two "Supplementary" angles when put together form, a "Straight" angle., , •, •, , "C" is for "Complementary" and "C" is for "Corner." Hence, you can, remember that two "Complementary" angles when put together form, a "Corner (right)" angle., , How to find the Supplement of an Angle?, , When the sum of two angles is equal to 180°, then we call that pair of angles, supplements, of each other. So, we know that the sum of two supplementary angles is 180 degrees, and, , 20/28

Page 22 :
each of them is said to be a "supplement" of the other. Thus, the supplement of an angle is, found by subtracting it from 180 degrees. This means the supplement of x° is (180 - x)°, For example, the supplement of 77° is obtained by subtracting it from 180°. Thus, its, supplement is (180-77)° = 103°., , Types of Triangles, The different types of triangles are classified according to the length of their sides and as, per the measure of the angles. The triangle is one of the most common shapes and is used, in construction for its rigidity and stable shape. Understanding these properties allows us to, apply the ideas in many real-world problems., What are the Different Types of Triangles?, , There are different types of triangles in math that can be distinguished based on their, sides and angles.

Page 23 :
Classifying Triangles, The characteristics of a triangle’s sides and angles are used to classify them. The different, types of triangles are as follows:, Types of Triangle, Based on Sides, , Types of Triangles, Based on Angles, , Equilateral Triangle, , Acute-Angled Triangle, (Acute Triangle), , Isosceles Triangle, , Right-Angled Triangle, (Right Triangle), , Scalene Triangle, , Obtuse-Angled Triangle, (Obtuse Triangle), , Types of Triangles Based on Sides, , On the basis of side lengths, the triangles are classified into the following, types:, •, , Equilateral Triangle: A triangle is considered to be an equilateral triangle when all three sides have the same, length., , •, , Isosceles triangle: When two sides of a triangle are equal or congruent, then it is called an isosceles triangle., , •, , Scalene triangle: When none of the sides of a triangle are equal, it is called a scalene triangle., , Types of Triangles Based on Angles, , On the basis of angles, triangles are classified into the following types:, •, , Acute Triangle: When all the angles of a triangle are acute, that is, they measure less than 90°, it is called an acuteangled triangle or acute triangle., , •, , Right Triangle: When one of the angles of a triangle is 90°, it is called a right-angled triangle or right triangle., , •, , Obtuse Triangle: When one of the angles of a triangle is an obtuse angle, that is, it measures greater than 90°, it is, called an obtuse-angled triangle or obtuse triangle., , 22/28

Page 24 :
Types of Triangle Based on Sides and Angles, , The different types of triangles are also classified according to their, sides and angles as follows:, •, , Equilateral or Equiangular Triangle: When all sides and angles of a triangle are equal, it is called an, equilateral or equiangular triangle., , •, , Isosceles Right Triangle: A triangle in which 2 sides are equal and one angle is 90° is called an, isosceles right triangle. So, in an isosceles right triangle, two sides and two acute angles are, congruent., , •, , Obtuse Isosceles Triangle: A triangle in which 2 sides are equal and one angle is an obtuse angle is, called an obtuse isosceles triangle., , •, , Acute Isosceles Triangle: A triangle in which all 3 angles are acute angles and 2 sides measure the, same is called an acute isosceles triangle., , •, , Right Scalene Triangle: A triangle in which any one of the angles is a right angle and all the 3 sides are, unequal, is called a right scalene triangle., , •, , Obtuse Scalene Triangle: A triangle with an obtuse angle with sides of different measures is called an, obtuse scalene triangle., , •, , Acute Scalene Triangle: A triangle that has 3 unequal sides and 3 acute angles is called an acute, scalene triangle., , ☛Important Notes:, Here is a list of a few points that should be remembered while, studying the types of triangles:, •, , In an equilateral triangle, each of the three internal angles is 60°., , •, , The three internal angles in a triangle always add up to 180°., , •, , All triangles have two acute angles., , •, , When all the sides and angles of a triangle are equal, it is called an equilateral or equiangular triangle.

Page 25 :
Circles, A circle is a simple closed curve and it is not a polygon. Below is the image of a circle., , Parts of a Circle, Centre: The center point of the entire circle. Point C is the centre of the circle., Radius: The line segment that connects the centre of circle to any point on the circle. OC is, the radius of the circle., Diameter: The line segment that connects two points on the circle and passes through the, centre of the circle. It is double the value of the radius. AB is the diameter of the circle., Chord: The line segment that connects two points on the circle but do not pass through the, centre of the circle. DE is the chord of the circle., A chord is any line segment touching the circle at two different points on its boundary., The longest chord in a circle is its diameter which passes through the center and divides, it into two equal parts., , Circumference: It is the distance around the circle., Arc: An arc is the part of circumference. DFE is the arc of the circle., Sector: Sector is an interior region bounded by pair of radii and arc on one side. OBC is a, sector of the circle where OB and OC are radius and BC is an arc., Segment: Segment is an interior region bounded by a chord and an arc., , 24/28

Page 27 :
Number of Diagonals Formula, , The number of diagonals in a polygon differs according to the type of polygon, based on the, number of sides. The formula that is used to find the number of diagonals in a polygon is,, Number of diagonals = n(n-3)/2; where 'n' represents the number of sides of the polygon., We can use this formula to find the number of diagonals of any polygon without actually, drawing them., , 26/28

Page 28 :
The following table shows the number of diagonals of some polygons which is calculated, using this formula., Shape, , Number of, sides (n), , Number of, Diagonals, , Triangle, , 3, , 3(3−3)/2 = 0, , Quadrilateral, , 4, , 4(4−3)/2 = 2, , Pentagon, , 5, , 5(5−3)/2 = 5, , Hexagon, , 6, , 6(6−3)/2 = 9, , Heptagon, , 7, , 7(7−3)/2 = 14, , Octagon, , 8, , 8(8−3)/2 = 20, , Nonagon, , 9, , 9(9−3)/2 = 27, , Decagon, , 10, , 10(10−3)/2 = 35, , Hendecagon, , 11, , 11(11−3)/2 = 44, , Dodecagon, , 12, , 12(12−3)/2 = 54, , Example: Find the number of diagonals in a decagon., Solution:, The number of sides in a decagon is n = 10. The number of diagonals of a decagon is, calculated using the formula:

Page 29 :
n(n-3)/2 = 10(10-3)/2 = 10(7)/2 = 70/2 = 35, Therefore, the number of diagonals of a decagon = 35, , 28/28