Study S1 Mathematics Maths - Basic Geometry - Geniebook

Basic Geometry

In this chapter, we will be discussing the below-mentioned topics in detail:

  • Basic geometrical concepts and notations 
  • Points, lines, planes
  • Types of angles
  • Complementary Vs Supplementary angles

 

  • Properties of angles formed by intersecting lines
  • Adjacent angles on a straight line
  • Angles at a point
  • Vertically opposite angles

 

A) i) Points

Description

Representation

  • Most basic geometrical object
  • Connecting 2 or more points forms other geometrical objects
  • A capital letter is used to label a point. Ex: A

             •

    A                                              •

                                                 B

 

ii) Line Segments

Description

Representation

  • Connecting 2 distinct points (called endpoints) forms a line segment
  • A line segment, AB, is labelled by its endpoints, A and B

•                                 •      

            A                                B

 

iii) Lines 

Description

Representation

  • A line is formed if a linesegment is extended indefinitely
  • A line has indefinite length but no breadth or thickness

              x                                        x     Line      A                                       B

 

iv) Rays 

Description

Representation

  • A ray is a line with only one endpoint.

 

v) Angles  

Description

Representation

  • An angle is formed by two rays, OA and OB, sharing the same endpoint, O.
  • O is the vertex, while OA and OB are sides of the angle. 
  • The angle is called angle AOB or angle BOA, written as \(\angle\)AOB or \(\angle\)BOA.

 

vi) Planes  

Description

Representation

  • A plane is a two-dimensional surface 
  • A plane has length and breadth, but no thickness
  • The floor is an example of a horizontal plane, and the wall is an example of a vertical plane.

 

B) Types Of Angles

Name

Definition

Illustration

Acute Angle

\(0^\circ < x^\circ < 90^\circ \)

More than \(0^\circ \)

Less than \(90^\circ \)

Right Angle

\(x^\circ = 90^\circ\)

Obtuse Angle

\(90^\circ < x^\circ < 180^\circ\) 

More than \(90^\circ \)

Less than \(180^\circ \)

Straight Angle

\(x^\circ  = 180^\circ\) 

Reflex Angle

\(180^\circ  < x^\circ  < 360^\circ  \)

More than \(180^\circ\) 

Less than \(360^\circ\) 

 

 

C) Complementary Angles vs Supplementary Angles

Complementary Angles

Supplementary Angles

Two angles are complementary if they add up to \(90^\circ \).

Two angles are supplementary if they add up to \(180^\circ \).

 

Let’s understand this with the help of some examples:

 

Question 1:

Angle POQ and angle QOR are supplementary. Angle POQ is three times the size of angle QOR. Find angle POQ.

 

  1. \(135^\circ\)                                                                         
  2. \(67.5^\circ\)        
  3. \(22.5^\circ\)                                                                
  4. \(45^\circ\)

 

Solution:

Let \(\angle QOR\) be \(x^\circ\) 

\(\begin{align*} \angle POQ &= 3x^\circ \\ \angle POQ + \angle QOR &= 180^\circ & \text { (supplementary } \angle)\\ 3x+x&=180\\ 4x&=180\\ x&=45 \\ \\ \angle POQ &=3(45)\\ &=135^\circ \end{align*}\)

Hence, the correct answer is Option (A).

 

D) Geometric Properties Of Points And Lines 

Illustration

Name

                                     

Collinear Points 

     Three points lie on the same line.

     

Intersecting Lines 

Two lines on a plane meet at one point.

Perpendicular Lines 

Two lines on a plane intersect each other at right angles.

Parallel Lines 

Two lines on a plane do not intersect at any point.

 

E) i) 1st Property Of Angles Formed By Intersecting Lines

 

\(\begin{equation} \angle a + \angle b + \angle c =180° \end{equation}\)

Property The sum of adjacent angles on a straight line is \(180°\).
Abbreviation adj. \(\angle s\) on a str. line.

 

ii) 2nd Property Of Angles Formed By Intersecting Lines

\(​\begin{align} \angle a + \angle b + \angle c + \angle d = 360° ​\end{align} ​\)

Property The sum of angles at a point is 360°.
Abbreviation \(\angle s\) at a pt.

 

iii) 3rd Property Of Angles Formed By Intersecting Lines

\(\begin{align*}​ \angle a &= \angle c  \\ \angle b &= \angle d​ \end{align*}\) 

Property  Vertically opposite angles are equal
Abbreviation  vert. opp. \(\angle s\).

 

Let’s understand this with the help of some examples:

 

Question 2:

i) In the figure, AOB and COD are straight lines. Find the value of p.

  1. \(\begin{align*}​ p=\frac {1}{13} \end{align*} \)
  2. \(\begin{align*}​ p=\frac {11}{13} \end{align*} \)
  3. \(\begin{align*}​ p=11 \end{align*} \)
  4. \(\begin{align*}​ p=1 \end{align*} \)
     

Solution: 

\(\begin{align*}​ \angle AOC &= \angle DOB & \text{(vert. opp. } \angle s\text{)} \\ \\ ​ 6p+6&=7p-5\\ 6+5&=7p-6p\\ 11&=p \\ \\ \therefore\qquad p &=11 \end{align*}\)

Hence, the correct answer is Option (C).

 

ii) In the figure, AOB and COD are straight lines. Find the value of q.

  1. \(\begin{align*}​ q &=104 \end{align*}\)
  2. \(\begin{align*}​ q=5\frac{3}{13} \end{align*}\)
  3. \(\begin{align*}​ q=8\frac{4}{13} \end{align*} \)
  4. \(\begin{align*}​ q=8 \end{align*}\)

 

Solution: 

\(\begin{align*}\small \angle{AOC} + \small \angle{COB} &= 180^\circ & \text { (adj angles on a str. line) } \\ 6p+6+13q+4&=180^\circ\\ \end{align*}\\\)

Putting values of \(p=11\)

\(\begin{align*} 6(11)+6+13q+4&=180^\circ\\ 13q+76&=180\\ 13q&=104\\ q&=8 \end{align*}\)

Hence, the correct answer is Option (D).

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