# 6.1 Triangles

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## Polygons

If you take some lines and join them such that the end point of the first line meets the starting point of the last line, you will get a polygon . Each line that makes up the polygon is known as a side . A polygon has interior angles. These are the angles that are inside the polygon. The number of sides of a polygon equals the number of interior angles. If a polygon has equal length sides and equal interior angles, then the polygon is called a regular polygon . Some examples of polygons are shown in [link] . Examples of polygons. They are all regular, except for the one marked *

## Triangles

A triangle is a three-sided polygon. There are four types of triangles: equilateral, isosceles, right-angled and scalene. The properties of these triangles are summarised in [link] .

 Name Diagram Properties equilateral All three sides are equal in length (denoted by the short lines drawn through all the sides of equal length) and all three angles are equal. isosceles Two sides are equal in length. The angles opposite the equal sides are equal. right-angled This triangle has one right angle. The side opposite this angle is called the hypotenuse . scalene (non-syllabus) All sides and angles are different.

If the corners of a triangle are denoted A, B and C - then we talk about $▵ABC$ .

## Investigation : sum of the angles in a triangle

1. Draw on a piece of paper a triangle of any size and shape
2. Cut it out and label the angles $\stackrel{^}{A}$ , $\stackrel{^}{B}$ and $\stackrel{^}{C}$ on both sides of the paper
3. Draw dotted lines as shown and cut along these lines to get three pieces of paper
4. Place them along your ruler as shown to see that $\stackrel{^}{A}+\stackrel{^}{B}+\stackrel{^}{C}={180}^{\circ }$

The sum of the angles in a triangle is 180 ${}^{\circ }$ . In any triangle, ∠ A + ∠ B + ∠ C = 180 ∘
Any exterior angle of a triangle is equal to the sum of the two opposite interior angles. An exterior angle is formed by extending any one of the sides. In any triangle, any exterior angle is equal to the sum of the two opposite interior angles.

## Congruent triangles

 Label Description Diagram RHS If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and the respective side of another triangle, then the triangles are congruent. SSS If three sides of a triangle are equal in length to the same sides of another triangle, then the two triangles are congruent SAS If two sides and the included angle of one triangle are equal to the same two sides and included angle of another triangle, then the two triangles are congruent. AAS If one side and two angles of one triangle are equal to the same one side and two angles of another triangle, then the two triangles are congruent. ## Similar triangles

 Description Diagram If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar. If all pairs of corresponding sides of two triangles are in proportion, then the triangles are similar. $\frac{x}{p}=\frac{y}{q}=\frac{z}{r}$

## The theorem of pythagoras

If $▵$ ABC is right-angled ( $\stackrel{^}{B}={90}^{\circ }$ ) then ${b}^{2}={a}^{2}+{c}^{2}$
Converse: If ${b}^{2}={a}^{2}+{c}^{2}$ , then $▵$ ABC is right-angled ( $\stackrel{^}{B}={90}^{\circ }$ ).

## Triangles

1. Calculate the unknown variables in each of the following figures. All lengths are in mm.
2. State whether or not the following pairs of triangles are congruent or not. Give reasons for your answers. If there is not enough information to make adescision, say why.

what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
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