Study S1 Mathematics Maths - Basic Geometry

Secondary 1

Maths

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

Most basic geometrical object
Connecting 2 or more points forms other geometrical objects
A capital letter is used to label a point. Ex: A

•

A •

ii) 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

~~• •~~

A B

iii) Lines

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

iv) Rays

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

v) 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.

vi) 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.

B) 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 \(360^\circ\)

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 \(180^\circ \).

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
	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.

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) 2^nd 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) 3^rd 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).

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