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 |
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• A • B |
ii) Line Segments
Description |
Representation |
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A B |
iii) Lines
Description |
Representation |
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iv) Rays
Description |
Representation |
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v) Angles
Description |
Representation |
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vi) Planes
Description |
Representation |
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B) Types Of Angles
Name |
Definition |
Illustration |
Acute Angle |
\(0^\circ < x^\circ < 90^\circ \) More than \(0^\circ \) Less than \(90^\circ \) |
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Right Angle |
\(x^\circ = 90^\circ\) |
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Obtuse Angle |
\(90^\circ < x^\circ < 180^\circ\) More than \(90^\circ \) Less than \(180^\circ \) |
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Straight Angle |
\(x^\circ = 180^\circ\) |
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Reflex Angle |
\(180^\circ < x^\circ < 360^\circ \) More than \(180^\circ\) Less than \(360^\circ\) |
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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 \(180^\circ \). |
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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.
- \(135^\circ\)
- \(67.5^\circ\)
- \(22.5^\circ\)
- \(45^\circ\)
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 |
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Collinear Points Three points lie on the same line. |
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Intersecting Lines Two lines on a plane meet at one point. |
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Perpendicular Lines Two lines on a plane intersect each other at right angles. |
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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.
- \(\begin{align*} p=\frac {1}{13} \end{align*} \)
- \(\begin{align*} p=\frac {11}{13} \end{align*} \)
- \(\begin{align*} p=11 \end{align*} \)
- \(\begin{align*} p=1 \end{align*} \)
Solution:
\(\begin{align*} \angle AOC &= \angle DOB & \text{(vert. opp. } \angle s\text{)} \\ \\ 6p+6&=7p-5\\ 6+5&=7p-6p\\ 11&=p \\ \\ \therefore\qquad p &=11 \end{align*}\)
Hence, the correct answer is Option (C).
ii) In the figure, AOB and COD are straight lines. Find the value of q.
- \(\begin{align*} q &=104 \end{align*}\)
- \(\begin{align*} q=5\frac{3}{13} \end{align*}\)
- \(\begin{align*} q=8\frac{4}{13} \end{align*} \)
- \(\begin{align*} q=8 \end{align*}\)
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).