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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 Figure 1.

Figure 1: Examples of polygons. They are all regular, except for the one marked *
Figure 1 (MG10C13_0221.png)

Triangles

A triangle is a three-sided polygon. Triangles are usually split into three categories: equilateral, isosceles, and scalene, depending on how many of the sides are of equal length. A fourth category, right-angled triangle (or simply 'right triangle') is used to refer to triangles with one right angle. Note that all right-angled triangles are also either isosceles (if the other two sides are equal) or scalene (it should be clear why you cannot have an equilateral right triangle!). The properties of these triangles are summarised in Table 1.

Table 1: Types of Triangles
Name Diagram Properties
equilateral
Figure 2
Figure 2 (MG10C13_023.png)
 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
Figure 3
Figure 3 (MG10C13_024.png)
 Two sides are equal in length. The angles opposite the equal sides are equal.
right-angled
Figure 4
Figure 4 (MG10C13_025.png)
 This triangle has one right angle. The side opposite this angle is called the hypotenuse.
scalene (non-syllabus)
Figure 5
Figure 5 (MG10C13_026.png)
 All sides and angles are different.

We use the notation ▵ABC▵ABC to refer to a triangle with corners labeled AA, BB, and CC.

Properties of Triangles

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 A^A^, B^B^ and C^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 A^+B^+C^=180∘A^+B^+C^=180∘

Figure 6
Figure 6 (MG10C13_027.png)
Figure 7
Figure 7 (MG10C13_028.png)

Tip:
The sum of the angles in a triangle is 180∘∘.
Figure 8: In any triangle, ∠A+∠B+∠C=180∘∠A+∠B+∠C=180∘
Figure 8 (MG10C13_029.png)
Tip:
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.
Figure 9: In any triangle, any exterior angle is equal to the sum of the two opposite interior angles.
Figure 9 (MG10C13_030.png)

Congruent Triangles

Two triangles are called congruent if one of them can be superimposed, that is moved on top of to exactly cover, the other. In other words, if both triangles have all of the same angles and sides, then they are called congruent. To decide whether two triangles are congruent, it is not necessary to check every side and angle. The following list describes various requirements that are sufficient to know when two triangles are congruent.

Table 2
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.
Figure 10
Figure 10 (MG10C13_031.png)
SSS If three sides of a triangle are equal in length to the same sides of another triangle, then the two triangles are congruent
Figure 11
Figure 11 (MG10C13_032.png)
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.
Figure 12
Figure 12 (MG10C13_033.png)
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.
Figure 13
Figure 13 (MG10C13_034.png)

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.

Table 3
Description Diagram
If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar.
Figure 14
Figure 14 (MG10C13_035.png)
If all pairs of corresponding sides of two triangles are in proportion, then the triangles are similar.
Figure 15
Figure 15 (MG10C13_036.png)
x p = y q = z r x p = y q = z r

The theorem of Pythagoras

Figure 16
Figure 16 (MG10C13_037.png)
If ▵▵ABC is right-angled (B^=90∘B^=90∘) then b2=a2+c2b2=a2+c2
Converse: If b2=a2+c2b2=a2+c2, then ▵▵ABC is right-angled (B^=90∘B^=90∘).

Exercise 1: Triangles

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

Figure 17
Figure 17 (triangle1.png)

Solution
  1. Step 1. Determine congruency:

    DEˆC=BAˆC=55°DEˆC=BAˆC=55°
    (angles in a triangle add up to 180°180°).

    ABˆC=CDˆE=90°ABˆC=CDˆE=90°
    (given)

    DE=AB=3DE=AB=3
    (given)

    ∴ Δ ABC ≡ Δ CDE ∴ΔABC≡ΔCDE
    (1)
  2. Step 2. Find the unknown variables:

    We use Pythagoras to find x:

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

    y=35°y=35°
    (angles in a triangle)

    z=5z=5
    (congruent triangles, AC=CEAC=CE)

Triangles
  1. Calculate the unknown variables in each of the following figures. All lengths are in mm.
    Figure 18
    Figure 18 (MG10C13_038.png)
    Click here for the solution
  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 a descision, say why.
    Figure 19
    Figure 19 (MG10C13_039.png)
    Click here for the solution

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 4: Table of some polygons and their number of sides.
Sides Name
5 pentagon
6 hexagon
7 heptagon
8 octagon
10 decagon
15 pentadecagon
Figure 20: Examples of other polygons.
Figure 20 (MG10C13_046.png)

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 ∘ A ^ = n - 2 n × 180 ∘
(3)

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

Exercise 2

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

Solution
  1. Step 1. Write down the number of sides in an octagon: An octagon has 8 sides.
  2. Step 2. Use the formula:
    A ^ = n - 2 n × 180 ∘ A ^ = 8 - 2 8 × 180 ∘ A ^ = 6 2 × 180 ∘ A ^ = 135 ∘ A ^ = n - 2 n × 180 ∘ A ^ = 8 - 2 8 × 180 ∘ A ^ = 6 2 × 180 ∘ A ^ = 135 ∘
    (4)

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∘0∘ and 90∘90∘
    • Right angle: An angle measuring 90∘90∘
    • Obtuse angle: An angle 90∘90∘ and 180∘180∘
    • Straight angle: An angle measuring 180∘180∘
    • Reflex angle: An angle 180∘180∘ and 360∘360∘
    • Revolution: An angle measuring 360∘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∘180∘

Exercises

  1. Find all the pairs of parallel lines in the following figures, giving reasons in each case.
    1. Figure 21
      Figure 21 (MG10C13_054.png)
    2. Figure 22
      Figure 22 (MG10C13_055.png)
    3. Figure 23
      Figure 23 (MG10C13_056.png)
    Click here for the solution
  2. Find angles aa, bb, cc and dd in each case, giving reasons.
    1. Figure 24
      Figure 24 (MG10C13_057.png)
    2. Figure 25
      Figure 25 (MG10C13_058.png)
    3. Figure 26
      Figure 26 (MG10C13_059.png)
    Click here for the solution
  3. Say which of the following pairs of triangles are congruent with reasons.
    1. Figure 27
      Figure 27 (MG10C13_060.png)
    2. Figure 28
      Figure 28 (MG10C13_061.png)
    3. Figure 29
      Figure 29 (MG10C13_062.png)
    4. Figure 30
      Figure 30 (MG10C13_063.png)
    Click here for the solution
  4. Identify the types of angles shown below (e.g. acute/obtuse etc):
    Figure 31
    Figure 31 (MG10C13_066.png)
    Click here for the solution
  5. Calculate the size of the third angle (x) in each of the diagrams below:
    Figure 32
    Figure 32 (MG10C13_067.png)
    Click here for the solution
  6. Name each of the shapes/polygons, state how many sides each has and whether it is regular (equiangular and equilateral) or not:
    Figure 33
    Figure 33 (MG10C13_068.png)
    Click here for the solution
  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.
    Click here for the solution
  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°
    Click here for the solution
  9. Using Pythagoras' theorem for right-angled triangles, calculate the length of x:
    Figure 34
    Figure 34 (MG10C13_070.png)
    Click here for the solution

Challenge Problem

  1. Using the figure below, show that the sum of the three angles in a triangle is 180∘∘. Line DEDE
    is parallel to BCBC.
    Figure 35
    Figure 35 (MG10C13_065.png)

    Click here for the solution

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