# Points, lines and angles  (Page 2/3)

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Angles are measured in degrees which is denoted by ${}^{\circ }$ , a small circle raised above the text in the same fashion as an exponent (or a superscript).

Angles can also be measured in radians. At high school level you will only use degrees, but if you decide to take maths at university you will learn about radians. Angle labelled as B ^ , ∠ C B A or ∠ A B C Examples of angles. A ^ = E ^ , even though the lines making up the angles are of different lengths.

## Measuring angles

The size of an angle does not depend on the length of the lines that are joined to make up the angle, but depends only on how both the lines are placed as can be seen in [link] . This means that the idea of length cannot be used to measure angles. An angle is a rotation around the vertex.

## Using a protractor

A protractor is a simple tool that is used to measure angles. A picture of a protractor is shown in [link] . Diagram of a protractor.

Method:

Using a protractor

1. Place the bottom line of the protractor along one line of the angle so that the other line of the angle points at the degree markings.
2. Move the protractor along the line so that the centre point on the protractor is at the vertex of the two lines that make up the angle.
3. Follow the second line until it meets the marking on the protractor and read off the angle. Make sure you start measuring at 0 ${}^{\circ }$ .

## Special angles

What is the smallest angle that can be drawn? The figure below shows two lines ( $CA$ and $AB$ ) making an angle at a common vertex $A$ . If line $CA$ is rotated around the common vertex $A$ , down towards line $AB$ , then the smallest angle that can be drawn occurs when the two lines are pointing in the same direction. This gives an angle of 0 ${}^{\circ }$ . This is shown in [link]

If line $CA$ is now swung upwards, any other angle can be obtained. If line $CA$ and line $AB$ point in opposite directions (the third case in [link] ) then this forms an angle of 180 ${}^{\circ }$ .

If three points $A$ , $B$ and $C$ lie on a straight line, then the angle between them is 180 ${}^{\circ }$ . Conversely, if the angle between three points is 180 ${}^{\circ }$ , then the points lie on a straight line.

An angle of 90 ${}^{\circ }$ is called a right angle . A right angle is half the size of the angle made by a straight line (180 ${}^{\circ }$ ). We say $CA$ is perpendicular to $AB$ or $CA\perp AB$ . An angle twice the size of a straight line is 360 ${}^{\circ }$ . An angle measuring 360 ${}^{\circ }$ looks identical to an angle of 0 ${}^{\circ }$ , except for the labelling. We call this a revolution . An angle of 90 ∘ is known as a right angle .

## Angles larger than 360 ${}^{\circ }$

All angles larger than 360 ${}^{\circ }$ also look like we have seen them before. If you are given an angle that is larger than 360 ${}^{\circ }$ , continue subtracting 360 ${}^{\circ }$ from the angle, until you get an answer that is between 0 ${}^{\circ }$ and 360 ${}^{\circ }$ . Angles that measure more than 360 ${}^{\circ }$ are largely for mathematical convenience.

• Acute angle : An angle $\ge {0}^{\circ }$ and $<{90}^{\circ }$ .
• Right angle : An angle measuring ${90}^{\circ }$ .
• Obtuse angle : An angle $>{90}^{\circ }$ and $<{180}^{\circ }$ .
• Straight angle : An angle measuring 180 ${}^{\circ }$ .
• Reflex angle : An angle $>{180}^{\circ }$ and $<{360}^{\circ }$ .
• Revolution : An angle measuring ${360}^{\circ }$ .

These are simply labels for angles in particular ranges, shown in [link] .

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
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