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Similar triangles

Two triangles are called similar if it is possible to proportionally shrink or stretch one of them to a triangle congruent to the other. Congruent triangles are similar triangles, but similar triangles are only congruent if they are the same size to begin with.

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.
x p = y q = z r

The theorem of pythagoras

If ABC is right-angled ( B ^ = 90 ) then b 2 = a 2 + c 2
Converse: If b 2 = a 2 + c 2 , then ABC is right-angled ( B ^ = 90 ).

In the following figure, determine if the two triangles are congruent, then use the result to help you find the unknown letters.

  1. D E ˆ C = B A ˆ C = 55 ° (angles in a triangle add up to 180 ° ).

    A B ˆ C = C D ˆ E = 90 ° (given)

    DE = AB = 3 (given)

  2. We use Pythagoras to find x:

    CE 2 = DE 2 + DC 2 5 2 = 3 2 + x 2 x 2 = 16 x = 4

    y = 35 ° (angles in a triangle)

    z = 5 (congruent triangles, AC = CE )

Got questions? Get instant answers now!


  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.


A quadrilateral is a four sided figure. There are some special quadrilaterals (trapezium, parallelogram, kite, rhombus, square, rectangle) which you will learn about in Geometry .

Other polygons

There are many other polygons, some of which are given in the table below.

Table of some polygons and their number of sides.
Sides Name
5 pentagon
6 hexagon
7 heptagon
8 octagon
10 decagon
15 pentadecagon
Examples of other polygons.

Angles of regular polygons

Polygons need not have all sides the same. When they do, they are called regular polygons. You can calculate the size of the interior angle of a regular polygon by using:

A ^ = n - 2 n × 180

where n is the number of sides and A ^ is any angle.

Find the size of the interior angles of a regular octagon.

  1. An octagon has 8 sides.
  2. A ^ = n - 2 n × 180 A ^ = 8 - 2 8 × 180 A ^ = 6 2 × 180 A ^ = 135
Got questions? Get instant answers now!


  • Make sure you know what the following terms mean: quadrilaterals, vertices, sides, angles, parallel lines, perpendicular lines,diagonals, bisectors and transversals.
  • The properties of triangles has been covered.
  • Congruency and similarity of triangles
  • Angles can be classified as acute, right, obtuse, straight, reflex or revolution
  • Theorem of Pythagoras which is used to calculate the lengths of sides of a right-angled triangle
  • Angles:
    • Acute angle: An angle 0 and 90
    • Right angle: An angle measuring 90
    • Obtuse angle: An angle 90 and 180
    • Straight angle: An angle measuring 180
    • Reflex angle: An angle 180 and 360
    • Revolution: An angle measuring 360
  • There are several properties of angles and some special names for these
  • There are four types of triangles: Equilateral, isoceles, right-angled and scalene
  • The angles in a triangle add up to 180


  1. Find all the pairs of parallel lines in the following figures, giving reasons in each case.
  2. Find angles a , b , c and d in each case, giving reasons.
  3. Say which of the following pairs of triangles are congruent with reasons.
  4. Identify the types of angles shown below (e.g. acute/obtuse etc):
  5. Calculate the size of the third angle (x) in each of the diagrams below:
  6. Name each of the shapes/polygons, state how many sides each has and whether it is regular (equiangular and equilateral) or not:
  7. Assess whether the following statements are true or false. If the statement is false, explain why:
    1. An angle is formed when two straight lines meet at a point.
    2. The smallest angle that can be drawn is 5°.
    3. An angle of 90° is called a square angle.
    4. Two angles whose sum is 180° are called supplementary angles.
    5. Two parallel lines will never intersect.
    6. A regular polygon has equal angles but not equal sides.
    7. An isoceles triangle has three equal sides.
    8. If three sides of a triangle are equal in length to the same sides of another triangle, then the two triangles are incongruent.
    9. If three pairs of corresponding angles in two triangles are equal, then the triangles are similar.
  8. Name the type of angle (e.g. acute/obtuse etc) based on it's size:
    1. 30°
    2. 47°
    3. 90°
    4. 91°
    5. 191°
    6. 360°
    7. 180°
  9. Using Pythagoras' theorem for right-angled triangles, calculate the length of x:

Challenge problem

  1. Using the figure below, show that the sum of the three angles in a triangle is 180 . Line D E is parallel to B C .

Questions & Answers

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anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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there is no specific books for beginners but there is book called principle of nanotechnology
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Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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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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