# Geometry basics: points, lines and angles

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## Introduction

The purpose of this chapter is to recap some of the ideas that you learned in geometry and trigonometry in earlier grades. You should feel comfortable with the work covered in this chapter before attempting to move onto the Grade 10 Geometry chapter , the Grade 10 Trigonometry chapter or the Grade 10 Analytical Geometry chapter . This chapter revises:

1. Terminology: vertices, sides, angles, parallel lines, perpendicular lines, diagonals, bisectors, transversals
2. Properties of triangles
3. Congruence
4. Classification of angles into acute, right, obtuse, straight, reflex or revolution
5. Theorem of Pythagoras which is used to calculate the lengths of sides of a right-angled triangle

## Points and lines

The two simplest objects in geometry are points and lines .

A point is a coordinate that marks a position in space (on a number line, on a plane or in three dimensions or even more) and is denoted by a dot. Points are usually labelled with a capital letter. Some examples of how points can be represented are shown in [link] .

A line is a continuous set of coordinates in space and can be thought of as being formed when many points are placed next to each other. Lines can be straight or curved, but are always continuous. This means that there are never any breaks in the lines (if there are, they would be distinct lines denoted separately). The endpoints of lines are labeled with capital letters. Examples of two lines are shown in [link] .

Lines are labelled according to the start point and end point. We call the line that starts at a point $A$ and ends at a point $B$ , $AB$ . Since the line from point $B$ to point $A$ is the same as the line from point $A$ to point $B$ , we have that $AB=BA$ .

When there is no ambiguity (which is the case throughout this text) the length of the line between points $A$ and $B$ is also denoted $AB$ , the same as the notation to refer to the line itself. So if we say $AB=CD$ we mean that the length of the line between $A$ and $B$ is equal to the length of the line between $C$ and $D$ .

Note: in higher mathematics, where there might be some ambiguity between when we want refer to the length of the line and when we just want to refer to the line itself, the notation $|AB|$ is usually used to refer to the length of the line. In this case, if one says $|AB|=|CD|$ , it means the lengths of the lines are the same, whereas if one says $AB=CD$ , it means that the two lines actually coincide (i.e. they are the same). Throughout this text, however, this notation will not be used, and $AB=CD$ ALWAYS implies that the lengths are the same.

A line is measured in units of length . Some common units of length are listed in [link] .

 Unit of Length Abbreviation kilometre km metre m centimetre cm millimetre mm

## Angles

An angle is formed when two straight lines meet at a point. The point at which two lines meet is known as a vertex . Angles are labelled with a $\phantom{\rule{0.277778em}{0ex}}\stackrel{^}{}\phantom{\rule{0.277778em}{0ex}}$ called a caret on a letter. For example, in [link] the angle is at $\stackrel{^}{B}$ . Angles can also be labelled according to the line segments that make up the angle. For example, in [link] the angle is made up when line segments $CB$ and $BA$ meet. So, the angle can be referred to as $\angle CBA$ or $\angle ABC$ or, if there is no ambiguity (i.e. there is only one angle at $B$ ) sometimes simply $\angle B$ . The $\angle$ symbol is a short method of writing angle in geometry.

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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