Basic Geometry
In this chapter, we will be discussing the belowmentioned 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

Points, Lines, Planes

Points
Description  Representation 



Line Segments
Description  Representation 



Lines
Description  Representation 



Rays
Description  Representation 



Angles
Description  Representation 



Planes
Description  Representation 



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 \(360^\circ\) 

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 \(\textit{POQ}\) and angle \(\textit{QOR}\) are supplementary. Angle \(\textit{POQ}\) is three times the size of angle \(\textit{QOR}\). Find angle \(\textit{POQ}\).
 \(135^\circ\)
 \(67.5^\circ\)
 \(22.5^\circ\)
 \(45^\circ\)
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).

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

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. 

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. 

3^{rd} 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:
 In the figure, AOB and COD are straight lines. Find the value of p.
 \(\displaystyle{p=\frac {1}{13}}\)
 \(\displaystyle{p=\frac {11}{13}}\)
 \(\begin{align*} p=11 \end{align*} \)
 \(\begin{align*} p=1 \end{align*} \)
Solution:
\(\begin{align*} \angle AOC &= \angle DOB & \text{(vert. opp. } \angle s\text{)} \\[2em] 6p+6&=7p5\\[2ex] 6+5&=7p6p\\[2ex] 11&=p \\[2ex] \therefore\qquad p &=11 \end{align*}\)
Hence, the correct answer is Option (C).
 In the figure, AOB and COD are straight lines. Find the value of q.
 \(\begin{align*} q &=104 \end{align*}\)
 \(\displaystyle{q=5\frac{3}{13}}\)
 \(\displaystyle{q=8\frac{4}{13}}\)
 \(\begin{align*} q=8 \end{align*}\)
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).