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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

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Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
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lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
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12, 17, 22.... 25th term
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12, 17, 22.... 25th term
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Carole
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AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
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Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
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find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
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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
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I'm not sure why it wrote it the other way
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I got X =-6
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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?
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hmm
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is it a question of log
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Commplementary angles
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A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
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Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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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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