# Geometry basics: points, lines and angles  (Page 3/3)

 Page 3 / 3

Once angles can be measured, they can then be compared. For example, all right angles are 90 ${}^{\circ }$ , therefore all right angles are equal and an obtuse angle will always be larger than an acute angle.

The following video summarizes what you have learnt so far about angles.

Note that for high school trigonometry you will be using degrees, not radians as stated in the video.

## Special angle pairs

In [link] , straight lines $AB$ and $CD$ intersect at point X, forming four angles: $\stackrel{^}{{X}_{1}}$ or $\angle BXD$ , $\stackrel{^}{{X}_{2}}$ or $\angle BXC$ , $\stackrel{^}{{X}_{3}}$ or $\angle CXA$ and $\stackrel{^}{{X}_{4}}$ or $\angle AXD$ .

The table summarises the special angle pairs that result.

 Special Angle Property Example adjacent angles share a common vertex and a common side $\left(\stackrel{^}{{X}_{1}},\stackrel{^}{{X}_{2}}\right)$ , $\left(\stackrel{^}{{X}_{2}},\stackrel{^}{{X}_{3}}\right)$ , $\left(\stackrel{^}{{X}_{3}},\stackrel{^}{{X}_{4}}\right)$ , $\left(\stackrel{^}{{X}_{4}},\stackrel{^}{{X}_{1}}\right)$ linear pair (adjacent angles on a straight line) adjacent angles formed by two intersecting straight lines that by definition add to 180 ${}^{\circ }$ $\stackrel{^}{{X}_{1}}+\stackrel{^}{{X}_{2}}={180}^{\circ }$ ; $\stackrel{^}{{X}_{2}}+\stackrel{^}{{X}_{3}}={180}^{\circ }$ ; $\stackrel{^}{{X}_{3}}+\stackrel{^}{{X}_{4}}={180}^{\circ }$ ; $\stackrel{^}{{X}_{4}}+\stackrel{^}{{X}_{1}}={180}^{\circ }$ opposite angles angles formed by two intersecting straight lines that share a vertex but do not share any sides $\stackrel{^}{{X}_{1}}=\stackrel{^}{{X}_{3}}$ ; $\stackrel{^}{{X}_{2}}=\stackrel{^}{{X}_{4}}$ supplementary angles two angles whose sum is 180 ${}^{\circ }$ complementary angles two angles whose sum is 90 ${}^{\circ }$
The opposite angles formed by the intersection of two straight lines are equal. Adjacent angles on a straight line are supplementary.

The following video summarises what you have learnt so far

## Parallel lines intersected by transversal lines

Two lines intersect if they cross each other at a point. For example, at a traffic intersection two or more streets intersect; the middle of the intersection is the common point between the streets.

Parallel lines are lines that never intersect. For example the tracks of a railway line are parallel (for convenience, sometimes mathematicians say they intersect at 'a point at infinity', i.e. an infinite distance away). We wouldn't want the tracks to intersect after as that would be catastrophic for the train!

All these lines are parallel to each other. Notice the pair of arrow symbols for parallel.

## Interesting fact

A section of the Australian National Railways Trans-Australian line is perhaps one of the longest pairs of man-made parallel lines.

Longest Railroad Straight (Source: www.guinnessworldrecords.com) The Australian National Railways Trans-Australian line over the Nullarbor Plain, is 478 km (297 miles) dead straight, from Mile 496, between Nurina and Loongana, Western Australia, to Mile 793, between Ooldea and Watson, South Australia.

A transversal of two or more lines is a line that intersects these lines. For example in [link] , $AB$ and $CD$ are two parallel lines and $EF$ is a transversal. We say $AB\parallel CD$ . The properties of the angles formed by these intersecting lines are summarised in the table below.

 Name of angle Definition Examples Notes interior angles the angles that lie inside the parallel lines in [link] $a$ , $b$ , $c$ and $d$ are interior angles the word interior means inside adjacent angles the angles share a common vertex point and line in [link] ( $a$ , $h$ ) are adjacent and so are ( $h$ , $g$ ); ( $g$ , $b$ ); ( $b$ , $a$ ) exterior angles the angles that lie outside the parallel lines in [link] $e$ , $f$ , $g$ and $h$ are exterior angles the word exterior means outside alternate interior angles the interior angles that lie on opposite sides of the transversal in [link] ( $a,c$ ) and ( $b$ , $d$ ) are pairs of alternate interior angles, $a=c$ , $b=d$ co-interior angles on the same side co-interior angles that lie on the same side of the transversal in [link] ( $a$ , $d$ ) and ( $b$ , $c$ ) are interior angles on the same side. $a+d={180}^{\circ }$ , $b+c={180}^{\circ }$ corresponding angles the angles on the same side of the transversal and the same side of the parallel lines in [link] $\left(a,e\right)$ , $\left(b,f\right)$ , $\left(c,g\right)$ and $\left(d,h\right)$ are pairs of corresponding angles. $a=e$ , $b=f$ , $c=g$ , $d=h$

The following video summarises what you have learnt so far

Euclid's Parallel line postulate. If a straight line falling across two other straight lines makes the two interior angles on the same side less than two right angles (180 ${}^{\circ }$ ), the two straight lines, if produced indefinitely, will meet on that side. This postulate can be used to prove many identities about the angles formed when two parallel lines are cut by a transversal.
1. If two parallel lines are intersected by a transversal, the sum of the co-interior angles on the same side of the transversal is 180 ${}^{\circ }$ .
2. If two parallel lines are intersected by a transversal, the alternate interior angles are equal.
3. If two parallel lines are intersected by a transversal, the corresponding angles are equal.
4. If two lines are intersected by a transversal such that any pair of co-interior angles on the same side is supplementary, then the two lines are parallel.
5. If two lines are intersected by a transversal such that a pair of alternate interior angles are equal, then the lines are parallel.
6. If two lines are intersected by a transversal such that a pair of alternate corresponding angles are equal, then the lines are parallel.

Find all the unknown angles in the following figure:

1. $\text{AB}\parallel \text{CD}$ . So $x={30}^{°}$ (alternate interior angles)
2. $\begin{array}{ccc}\hfill 160+y& =& 180\hfill \\ \hfill y& =& {20}^{°}\hfill \end{array}$
(co-interior angles on the same side)

Determine if there are any parallel lines in the following figure:

1. Line EF cannot be parallel to either AB or CD since it cuts both these lines. Lines AB and CD may be parallel.
2. We can show that two lines are parallel if we can find one of the pairs of special angles. We know that ${\stackrel{ˆ}{E}}_{2}={25}^{°}$ (opposite angles). And then we note that
$\begin{array}{ccc}\hfill {\stackrel{ˆ}{E}}_{2}& =& {\stackrel{ˆ}{F}}_{4}\hfill \\ & =& {25}^{°}\hfill \end{array}$
So we have shown that $\text{AB}\parallel \text{CD}$ (corresponding angles)

## Angles

1. Use adjacent, corresponding, co-interior and alternate angles to fill in all the angles labeled with letters in the diagram below:
2. Find all the unknown angles in the figure below:
3. Find the value of $x$ in the figure below:
4. Determine whether there are pairs of parallel lines in the following figures.
5. If AB is parallel to CD and AB is parallel to EF, prove that CD is parallel to EF:

The following video shows some problems with their solutions

how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
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
I'm interested in nanotube
Uday
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
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!