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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)

    Δ ABC Δ CDE
  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 )

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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.

Quadrilaterals

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
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Summary

  • 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

Exercises

  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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
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
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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J, combine like terms 7x-4y
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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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
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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.
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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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