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Introduction

The purpose of this chapter is to recap some of the ideas that you learned in geometry and trigonometry in earlier grades. You should feel comfortable with the work covered in this chapter before attempting to move onto the Grade 10 Geometry chapter, the Grade 10 Trigonometry chapter or the Grade 10 Analytical Geometry chapter. This chapter revises:

  1. Terminology: vertices, sides, angles, parallel lines, perpendicular lines, diagonals, bisectors, transversals
  2. Properties of triangles
  3. Congruence
  4. Classification of angles into acute, right, obtuse, straight, reflex or revolution
  5. Theorem of Pythagoras which is used to calculate the lengths of sides of a right-angled triangle

Points and Lines

The two simplest objects in geometry are points and lines.

A point is a coordinate that marks a position in space (on a number line, on a plane or in three dimensions or even more) and is denoted by a dot. Points are usually labelled with a capital letter. Some examples of how points can be represented are shown in Figure 1.

A line is a continuous set of coordinates in space and can be thought of as being formed when many points are placed next to each other. Lines can be straight or curved, but are always continuous. This means that there are never any breaks in the lines (if there are, they would be distinct lines denoted separately). The endpoints of lines are labeled with capital letters. Examples of two lines are shown in Figure 1.

Figure 1: Examples of some points (labelled PP, QQ, RR and SS
) and some lines (labelled BCBC and DEDE
).
Figure 1 (MG10C13_001.png)

Lines are labelled according to the start point and end point. We call the line that starts at a point AA and ends at a point BB, ABAB. Since the line from point BB to point AA is the same as the line from point AA to point BB, we have that AB=BAAB=BA.

When there is no ambiguity (which is the case throughout this text) the length of the line between points AA and BB is also denoted ABAB
, the same as the notation to refer to the line itself. So if we say AB=CDAB=CD
we mean that the length of the line between AA and BB is equal to the length of the line between CC and DD.

Note: in higher mathematics, where there might be some ambiguity between when we want refer to the length of the line and when we just want to refer to the line itself, the notation |AB||AB|
is usually used to refer to the length of the line. In this case, if one says |AB|=|CD||AB|=|CD|, it means the lengths of the lines are the same, whereas if one says AB=CDAB=CD, it means that the two lines actually coincide (i.e. they are the same). Throughout this text, however, this notation will not be used, and AB=CDAB=CD ALWAYS implies that the lengths are the same.

A line is measured in units of length. Some common units of length are listed in Table 1.

Table 1: Some common units of length and their abbreviations.
Unit of Length Abbreviation
kilometre km
metre m
centimetre cm
millimetre mm

Angles

An angle is formed when two straight lines meet at a point. The point at which two lines meet is known as a vertex. Angles are labelled with a ^^ called a caret on a letter. For example, in Figure 2 the angle is at B^B^. Angles can also be labelled according to the line segments that make up the angle. For example, in Figure 2 the angle is made up when line segments CBCB and BABA meet. So, the angle can be referred to as ∠CBA∠CBA
or ∠ABC∠ABC
or, if there is no ambiguity (i.e. there is only one angle at BB) sometimes simply ∠B∠B. The ∠∠ symbol is a short method of writing angle in geometry.

Angles are measured in degrees which is denoted by ∘∘, a small circle raised above the text in the same fashion as an exponent (or a superscript).

Note:

Angles can also be measured in radians. At high school level you will only use degrees, but if you decide to take maths at university you will learn about radians.

Figure 2: Angle labelled as B^B^, ∠CBA∠CBA
or ∠ABC∠ABC
Figure 2 (MG10C13_002.png)
Figure 3: Examples of angles. A^=E^A^=E^, even though the lines making up the angles are of different lengths.
Figure 3 (MG10C13_003.png)

Measuring angles

The size of an angle does not depend on the length of the lines that are joined to make up the angle, but depends only on how both the lines are placed as can be seen in Figure 3. This means that the idea of length cannot be used to measure angles. An angle is a rotation around the vertex.

Using a Protractor

A protractor is a simple tool that is used to measure angles. A picture of a protractor is shown in Figure 4.

Figure 4: Diagram of a protractor.
Figure 4 (MG10C13_004.png)

Method:

Using a protractor

  1. Place the bottom line of the protractor along one line of the angle so that the other line of the angle points at the degree markings.
  2. Move the protractor along the line so that the centre point on the protractor is at the vertex of the two lines that make up the angle.
  3. Follow the second line until it meets the marking on the protractor and read off the angle. Make sure you start measuring at 0∘∘.
Measuring Angles : Use a protractor to measure the following angles:

Figure 5
Figure 5 (MG10C13_005.png)

Special Angles

What is the smallest angle that can be drawn? The figure below shows two lines (CACA and ABAB) making an angle at a common vertex AA. If line CACA is rotated around the common vertex AA, down towards line ABAB, then the smallest angle that can be drawn occurs when the two lines are pointing in the same direction. This gives an angle of 0∘∘. This is shown in Figure 6

Figure 6
Figure 6 (MG10C13_006.png)

If line CACA is now swung upwards, any other angle can be obtained. If line CACA and line ABAB point in opposite directions (the third case in Figure 6) then this forms an angle of 180∘∘.

Tip:

If three points AA, BB and CC lie on a straight line, then the angle between them is 180∘∘. Conversely, if the angle between three points is 180∘∘, then the points lie on a straight line.

An angle of 90∘∘ is called a right angle. A right angle is half the size of the angle made by a straight line (180∘∘). We say CACA is perpendicular to ABAB or CA⊥ABCA⊥AB
. An angle twice the size of a straight line is 360∘∘. An angle measuring 360∘∘ looks identical to an angle of 0∘∘, except for the labelling. We call this a revolution.

Figure 7: An angle of 90∘∘ is known as a right angle.
Figure 7 (MG10C13_007.png)

Angles larger than 360∘∘

All angles larger than 360∘∘ also look like we have seen them before. If you are given an angle that is larger than 360∘∘, continue subtracting 360∘∘ from the angle, until you get an answer that is between 0∘∘and 360∘∘. Angles that measure more than 360∘∘ are largely for mathematical convenience.

Tip:

  • 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∘∘.
  • Reflex angle: An angle >180∘>180∘ and <360∘<360∘.
  • Revolution: An angle measuring 360∘360∘.

These are simply labels for angles in particular ranges, shown in Figure 8.

Figure 8: Three types of angles defined according to their ranges.
Figure 8 (MG10C13_008.png)

Once angles can be measured, they can then be compared. For example, all right angles are 90∘∘, therefore all right angles are equal and an obtuse angle will always be larger than an acute angle.

The following video summarizes what you have learnt so far about angles.

Figure 9
Khan Academy video on angles - 1
Note that for high school trigonometry you will be using degrees, not radians as stated in the video.

Special Angle Pairs

In Figure 10, straight lines ABAB and CDCD intersect at point X, forming four angles: X1^X1^ or ∠BXD∠BXD
, X2^X2^
or ∠BXC∠BXC
, X3^X3^
or ∠CXA∠CXA
and X4^X4^
or ∠AXD∠AXD
.

Figure 10: Two intersecting straight lines with vertical angles X1^,X3^X1^,X3^ and X2^,X4^X2^,X4^.
Figure 10 (MG10C13_009.png)

The table summarises the special angle pairs that result.

Table 2
Special Angle Property Example
adjacent angles share a common vertex and a common side (X1^,X2^)(X1^,X2^), (X2^,X3^)(X2^,X3^), (X3^,X4^)(X3^,X4^), (X4^,X1^)(X4^,X1^)
linear pair (adjacent angles on a straight line) adjacent angles formed by two intersecting straight lines that by definition add to 180∘∘ X 1 ^ + X 2 ^ = 180 ∘ X 1 ^ + X 2 ^ = 180 ∘ ; X 2 ^ + X 3 ^ = 180 ∘ X 2 ^ + X 3 ^ = 180 ∘ ; X 3 ^ + X 4 ^ = 180 ∘ X 3 ^ + X 4 ^ = 180 ∘ ; X 4 ^ + X 1 ^ = 180 ∘ X 4 ^ + X 1 ^ = 180 ∘
opposite angles angles formed by two intersecting straight lines that share a vertex but do not share any sides X 1 ^ = X 3 ^ X 1 ^ = X 3 ^ ; X 2 ^ = X 4 ^ X 2 ^ = X 4 ^
supplementary angles two angles whose sum is 180∘∘
complementary angles two angles whose sum is 90∘∘

Tip:

The opposite angles formed by the intersection of two straight lines are equal. Adjacent angles on a straight line are supplementary.

The following video summarises what you have learnt so far

Figure 11
Khan Academy video on angles - 2

Parallel Lines intersected by Transversal Lines

Two lines intersect if they cross each other at a point. For example, at a traffic intersection two or more streets intersect; the middle of the intersection is the common point between the streets.

Parallel lines are lines that never intersect. For example the tracks of a railway line are parallel (for convenience, sometimes mathematicians say they intersect at 'a point at infinity', i.e. an infinite distance away). We wouldn't want the tracks to intersect after as that would be catastrophic for the train!

Figure 12
Figure 12 (MG10C13_010.png)

All these lines are parallel to each other. Notice the pair of arrow symbols for parallel.

Note: Interesting Fact :

A section of the Australian National Railways Trans-Australian line is perhaps one of the longest pairs of man-made parallel lines.

Longest Railroad Straight (Source: www.guinnessworldrecords.com) The Australian National Railways Trans-Australian line over the Nullarbor Plain, is 478 km (297 miles) dead straight, from Mile 496, between Nurina and Loongana, Western Australia, to Mile 793, between Ooldea and Watson, South Australia.

A transversal of two or more lines is a line that intersects these lines. For example in Figure 13, ABAB and CDCD are two parallel lines and EFEF is a transversal. We say AB∥CDAB∥CD. The properties of the angles formed by these intersecting lines are summarised in the table below.

Figure 13: Parallel lines intersected by a transversal
Figure 13 (MG10C13_011.png)
Table 3
Name of angle Definition Examples Notes
interior angles the angles that lie inside the parallel lines in Figure 13 aa, bb, cc and dd are interior angles the word interior means inside
adjacent angles the angles share a common vertex point and line in Figure 13 (aa, hh) are adjacent and so are (hh, gg); (gg, bb); (bb, aa)  
exterior angles the angles that lie outside the parallel lines in Figure 13 ee, ff, gg and hh are exterior angles the word exterior means outside
alternate interior angles the interior angles that lie on opposite sides of the transversal in Figure 13 (a,ca,c) and (bb,dd) are pairs of alternate interior angles, a=ca=c, b=db=d
Figure 14
Figure 14 (MG10C13_012.png)
co-interior angles on the same side co-interior angles that lie on the same side of the transversal in Figure 13 (aa,dd) and (bb,cc) are interior angles on the same side. a+d=180∘a+d=180∘, b+c=180∘b+c=180∘
Figure 15
Figure 15 (MG10C13_013.png)
corresponding angles the angles on the same side of the transversal and the same side of the parallel lines in Figure 13 (a,e)(a,e), (b,f)(b,f), (c,g)(c,g) and (d,h)(d,h) are pairs of corresponding angles. a=ea=e, b=fb=f, c=gc=g, d=hd=h
Figure 16
Figure 16 (MG10C13_014.png)

The following video summarises what you have learnt so far

Figure 17
Khan Academy video on angles - 3

Note:

Euclid's Parallel line postulate. If a straight line falling across two other straight lines makes the two interior angles on the same side less than two right angles (180∘∘), the two straight lines, if produced indefinitely, will meet on that side. This postulate can be used to prove many identities about the angles formed when two parallel lines are cut by a transversal.

Tip:

  1. If two parallel lines are intersected by a transversal, the sum of the co-interior angles on the same side of the transversal is 180∘∘.
  2. If two parallel lines are intersected by a transversal, the alternate interior angles are equal.
  3. If two parallel lines are intersected by a transversal, the corresponding angles are equal.
  4. If two lines are intersected by a transversal such that any pair of co-interior angles on the same side is supplementary, then the two lines are parallel.
  5. If two lines are intersected by a transversal such that a pair of alternate interior angles are equal, then the lines are parallel.
  6. If two lines are intersected by a transversal such that a pair of alternate corresponding angles are equal, then the lines are parallel.

Exercise 1: Finding angles

Find all the unknown angles in the following figure:

Figure 18
Figure 18 (angle1.png)

[ Show Solution ]

Exercise 2: Parallel lines

Determine if there are any parallel lines in the following figure:

Figure 19
Figure 19 (angle2.png)

[ Show Solution ]

Angles

  1. Use adjacent, corresponding, co-interior and alternate angles to fill in all the angles labeled with letters in the diagram below:
    Figure 20
    Figure 20 (MG10C13_015.png)
    Click here for the solution
  2. Find all the unknown angles in the figure below:
    Figure 21
    Figure 21 (MG10C13_016.png)
    Click here for the solution
  3. Find the value of xx in the figure below:
    Figure 22
    Figure 22 (MG10C13_017.png)
    Click here for the solution
  4. Determine whether there are pairs of parallel lines in the following figures.
    1. Figure 23
      Figure 23 (MG10C13_018.png)
    2. Figure 24
      Figure 24 (MG10C13_019.png)
    3. Figure 25
      Figure 25 (MG10C13_020.png)
    Click here for the solution
  5. If AB is parallel to CD and AB is parallel to EF, prove that CD is parallel to EF:
    Figure 26
    Figure 26 (MG10C13_021.png)
    Click here for the solution

The following video shows some problems with their solutions

Figure 27
Khan Academy video on angles - 4

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 28.

Figure 28: Examples of polygons. They are all regular, except for the one marked *
Figure 28 (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 4.

Table 4: Types of Triangles
Name Diagram Properties
equilateral
Figure 29
Figure 29 (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 30
Figure 30 (MG10C13_024.png)
 Two sides are equal in length. The angles opposite the equal sides are equal.
right-angled
Figure 31
Figure 31 (MG10C13_025.png)
 This triangle has one right angle. The side opposite this angle is called the hypotenuse.
scalene (non-syllabus)
Figure 32
Figure 32 (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 33
Figure 33 (MG10C13_027.png)
Figure 34
Figure 34 (MG10C13_028.png)

Tip:
The sum of the angles in a triangle is 180∘∘.
Figure 35: In any triangle, ∠A+∠B+∠C=180∘∠A+∠B+∠C=180∘
Figure 35 (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 36: In any triangle, any exterior angle is equal to the sum of the two opposite interior angles.
Figure 36 (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 5
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 37
Figure 37 (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 38
Figure 38 (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 39
Figure 39 (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 40
Figure 40 (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 6
Description Diagram
If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar.
Figure 41
Figure 41 (MG10C13_035.png)
If all pairs of corresponding sides of two triangles are in proportion, then the triangles are similar.
Figure 42
Figure 42 (MG10C13_036.png)
x p = y q = z r x p = y q = z r

The theorem of Pythagoras

Figure 43
Figure 43 (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 3: Triangles

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

Figure 44
Figure 44 (triangle1.png)

[ Show Solution ]
Triangles
  1. Calculate the unknown variables in each of the following figures. All lengths are in mm.
    Figure 45
    Figure 45 (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 46
    Figure 46 (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 7: Table of some polygons and their number of sides.
Sides Name
5 pentagon
6 hexagon
7 heptagon
8 octagon
10 decagon
15 pentadecagon
Figure 47: Examples of other polygons.
Figure 47 (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 ∘
(5)

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

Exercise 4

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

[ Show Solution ]

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 48
      Figure 48 (MG10C13_054.png)
    2. Figure 49
      Figure 49 (MG10C13_055.png)
    3. Figure 50
      Figure 50 (MG10C13_056.png)
    Click here for the solution
  2. Find angles aa, bb, cc and dd in each case, giving reasons.
    1. Figure 51
      Figure 51 (MG10C13_057.png)
    2. Figure 52
      Figure 52 (MG10C13_058.png)
    3. Figure 53
      Figure 53 (MG10C13_059.png)
    Click here for the solution
  3. Say which of the following pairs of triangles are congruent with reasons.
    1. Figure 54
      Figure 54 (MG10C13_060.png)
    2. Figure 55
      Figure 55 (MG10C13_061.png)
    3. Figure 56
      Figure 56 (MG10C13_062.png)
    4. Figure 57
      Figure 57 (MG10C13_063.png)
    Click here for the solution
  4. Identify the types of angles shown below (e.g. acute/obtuse etc):
    Figure 58
    Figure 58 (MG10C13_066.png)
    Click here for the solution
  5. Calculate the size of the third angle (x) in each of the diagrams below:
    Figure 59
    Figure 59 (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 60
    Figure 60 (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 61
    Figure 61 (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 62
    Figure 62 (MG10C13_065.png)

    Click here for the solution

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