# 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

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
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
1. ### Lines

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

Description Representation
• A ray is a line with only one endpoint.
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 $\angle BOA.$
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.
1. ### 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
$360^\circ$

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 $180^\circ$.

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

$$$\angle a + \angle b + \angle c =180°$$$

Property The sum of adjacent angles on a straight line is $180°$. 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°. $\angle s$ 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 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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