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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
     

Basic Geometrical Concepts And Notations 

  1. Points, Lines, Planes

  1. 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
Points Representation
  1. 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
Line Segments
  1. Lines 

Description Representation
  • A line is formed if a linesegment is extended indefinitely
  • A line has indefinite length but no breadth or thickness
Lines 
  1. Rays 

Description Representation
  • A ray is a line with only one endpoint.
Rays
  1. 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 \(\angle BOA.\)
Angles
  1. 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.
Planes
  1. Types Of Angles

Name Definition Illustration
Acute Angle \(0^\circ < x^\circ < 90^\circ \)
More than \(0^\circ \)
Less than \(90^\circ \)
Acute Angle
Right Angle \(x^\circ = 90^\circ\) Right Angle
Obtuse Angle \(90^\circ < x^\circ < 180^\circ\)
More than \(90^\circ \)
Less than \(180^\circ \)
Obtuse Angle
Straight Angle \(x^\circ  = 180^\circ\) Straight Angle
Reflex Angle \(180^\circ  < x^\circ  < 360^\circ  \)
More than \(180^\circ\)
Less than 
\(360^\circ\)
Reflex Angle

 

  1. 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 \(180^\circ \).
Complementary Angles Supplementary Angles

 

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

Question 1:

Angle \(\textit{POQ}\) and angle \(\textit{QOR}\) are supplementary. Angle \(\textit{POQ}\) is three times the size of angle \(\textit{QOR}\). Find angle \(\textit{POQ}\).

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

Solution:

Let \(\angle QOR\) be \(x^\circ\) 

\(\begin{align*} \angle POQ &= 3x^\circ \\[2ex] \angle POQ + \angle QOR &= 180^\circ & \text { (supplementary } \angle)\\[2ex] 3x+x&=180\\[2ex] 4x&=180\\[2ex] x&=45 \\[2em] \angle POQ &=3(45)\\[2ex] &=135^\circ \end{align*}\)

Hence, the correct answer is Option (A).

 

  1. 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.

 

Properties Of Angles Formed By Intersecting Lines

  1. 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.

 

  1. 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.

 

  1. 3rd Property Of Angles Formed By Intersecting Lines

\(\begin{align*}​ \angle a &= \angle c  \\[2ex] \angle b &= \angle d​ \end{align*}\)

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

 

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

 

Question 2:

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

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

Solution: 

\(\begin{align*}​ \angle AOC &= \angle DOB & \text{(vert. opp. } \angle s\text{)} \\[2em] 6p+6&=7p-5\\[2ex] 6+5&=7p-6p\\[2ex] 11&=p \\[2ex] \therefore\qquad p &=11 \end{align*}\)

Hence, the correct answer is Option (C).

 

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

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

Solution: 

\(\begin{align*} \small \angle{AOC} + \small \angle{COB} &= 180^\circ & \text { (adj angles on a str. line) } \\[2ex] 6p+6+13q+4&=180^\circ \end{align*}\\\)

Putting values of \(p=11\)

\(\begin{align*} 6(11)+6+13q+4&=180^\circ\\[2ex] 13q+76&=180\\[2ex] 13q&=104\\[2ex] q&=8 \end{align*}\)

Hence, the correct answer is Option (D).

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