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

2. 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

3. Lines 

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

4. Rays 

Description	Representation
A ray is a line with only one endpoint.

5. 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 \(\angle BOA.\)

6. 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.

2. 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 \(360^\circ\)

3. 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 \(\textit{POQ}\).

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).

4. 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

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

2. 2nd Property Of Angles Formed By Intersecting Lines

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

3. 3^rd Property Of Angles Formed By Intersecting Lines

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

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).

2. 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).