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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 or the Grade 10 Trigonometry Chapter. This chapter revises:

  1. Terminology: quadrilaterals, vertices, sides, angles, parallel lines, perpendicular lines, diagonals, bisectors, transversals
  2. Similarities and differences between quadrilaterals
  3. Properties of triangles and quadrilaterals
  4. Congruence
  5. Classification of angles into acute, right, obtuse, straight, reflex or revolution
  6. 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. The endpoints of lines are labelled 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.

The length of the line between points AA and BB is ABAB

 
. 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.

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 CBACBA
 
or ABCABC
 
. 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^, CBACBA
 
or ABCABC
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 CAABCAAB

 
. 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 0and 360. Angles that measure more than 360 are largely for mathematical convenience.

Tip:

  • Acute angle: An angle 00 and <90<90.
  • Right angle: An angle measuring 9090.
  • 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 360360.

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. Radians are simply another way to measure angles. At university level you will learn about radians.

Special Angle Pairs

In Figure 10, straight lines ABAB and CDCD intersect at point X, forming four angles: X1^X1^ or BXDBXD

 
, X2^X2^
 
or BXCBXC
 
, X3^X3^
 
or CXACXA
 
and X4^X4^
 
or AXDAXD
 
.

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. We wouldn't want the tracks to intersect 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 ABCDABCD. 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=180a+d=180, b+c=180b+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 on two 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.

Angles

  1. Use adjacent, corresponding, co-interior and alternate angles to fill in all the angles labeled with letters in the diagram below:
    Figure 18
    Figure 18 (MG10C13_015.png)
    Click here for the solution
  2. Find all the unknown angles in the figure below:
    Figure 19
    Figure 19 (MG10C13_016.png)
    Click here for the solution
  3. Find the value of xx in the figure below:
    Figure 20
    Figure 20 (MG10C13_017.png)
    Click here for the solution
  4. Determine whether there are pairs of parallel lines in the following figures.
    1. Figure 21
      Figure 21 (MG10C13_018.png)
    2. Figure 22
      Figure 22 (MG10C13_019.png)
    3. Figure 23
      Figure 23 (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 24
    Figure 24 (MG10C13_021.png)
    Click here for the solution

The following video shows some problems with their solutions

Figure 25
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 26.

Figure 26: Examples of polygons. They are all regular, except for the one marked *
Figure 26 (MG10C13_0221.png)

Triangles

A triangle is a three-sided polygon. There are four types of triangles: equilateral, isosceles, right-angled and scalene. The properties of these triangles are summarised in Table 4.

Table 4: Types of Triangles
Name Diagram Properties
equilateral
Figure 27
Figure 27 (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 28
Figure 28 (MG10C13_024.png)
Two sides are equal in length. The angles opposite the equal sides are equal.
right-angled
Figure 29
Figure 29 (MG10C13_025.png)
This triangle has one right angle. The side opposite this angle is called the hypotenuse.
scalene (non-syllabus)
Figure 30
Figure 30 (MG10C13_026.png)
All sides and angles are different.

If the corners of a triangle are denoted A, B and C - then we talk about ABCABC.

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^=180A^+B^+C^=180

Figure 31
Figure 31 (MG10C13_027.png)
Figure 32
Figure 32 (MG10C13_028.png)

Tip:
The sum of the angles in a triangle is 180.
Figure 33: In any triangle, A+B+C=180A+B+C=180
Figure 33 (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 34: In any triangle, any exterior angle is equal to the sum of the two opposite interior angles.
Figure 34 (MG10C13_030.png)

Congruent Triangles

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 35
Figure 35 (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 36
Figure 36 (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 37
Figure 37 (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 38
Figure 38 (MG10C13_034.png)

Similar Triangles

Table 6
Description Diagram
If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar.
Figure 39
Figure 39 (MG10C13_035.png)
If all pairs of corresponding sides of two triangles are in proportion, then the triangles are similar.
Figure 40
Figure 40 (MG10C13_036.png)
x p = y q = z r x p = y q = z r

The theorem of Pythagoras

Figure 41
Figure 41 (MG10C13_037.png)
If ABC is right-angled (B^=90B^=90) then b2=a2+c2b2=a2+c2
Converse: If b2=a2+c2b2=a2+c2, then ABC is right-angled (B^=90B^=90).

Triangles
  1. Calculate the unknown variables in each of the following figures. All lengths are in mm.
    Figure 42
    Figure 42 (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 43
    Figure 43 (MG10C13_039.png)
    Click here for the solution

Quadrilaterals

A quadrilateral is any polygon with four sides. The basic quadrilaterals are the trapezium, parallelogram, rectangle, rhombus, square and kite.

Table 7: Examples of quadrilaterals.
Name of quadrilateral Figure
trapezium Figure 44
parallelogram Figure 45
rectangle Figure 46
rhombus Figure 47
square Figure 48
kite Figure 49

Trapezium

A trapezium is a quadrilateral with one pair of parallel opposite sides. It may also be called a trapezoid. A special type of trapezium is the isosceles trapezium, where one pair of opposite sides is parallel, the other pair of sides is equal in length and the angles at the ends of each parallel side are equal. An isosceles trapezium has one line of symmetry and its diagonals are equal in length.

Figure 44: Examples of trapeziums.
Figure 44 (MG10C13_040.png)

Parallelogram

A trapezium with both sets of opposite sides parallel is called a parallelogram. A summary of the properties of a parallelogram is:

  • Both pairs of opposite sides are parallel.
  • Both pairs of opposite sides are equal in length.
  • Both pairs of opposite angles are equal.
  • Both diagonals bisect each other (i.e. they cut each other in half).
Figure 45: An example of a parallelogram.
Figure 45 (MG10C13_041.png)

Rectangle

A rectangle is a parallelogram that has all four angles equal to 9090. A summary of the properties of a rectangle is:

  • Both pairs of opposite sides are parallel.
  • Both pairs of opposite sides are of equal length.
  • Both diagonals bisect each other.
  • Diagonals are equal in length.
  • All angles at the corners are right angles.
Figure 46: Example of a rectangle.
Figure 46 (MG10C13_042.png)

Rhombus

A rhombus is a parallelogram that has all four sides of equal length. A summary of the properties of a rhombus is:

  • Both pairs of opposite sides are parallel.
  • All sides are equal in length.
  • Both pairs of opposite angles are equal.
  • Both diagonals bisect each other at 9090.
  • Diagonals of a rhombus bisect both pairs of opposite angles.
Figure 47: An example of a rhombus. A rhombus is a parallelogram with all sides equal.
Figure 47 (MG10C13_043.png)

Square

A square is a rhombus that has all four angles equal to 90.

A summary of the properties of a square is:

  • Both pairs of opposite sides are parallel.
  • All sides are equal in length.
  • All angles are equal to 9090.
  • Both pairs of opposite angles are equal.
  • Both diagonals bisect each other at 9090.
  • Diagonals are equal in length.
  • Diagonals bisect both pairs of opposite angles (ie. all 4545).
Figure 48: An example of a square. A square is a rhombus with all angles equal to 90.
Figure 48 (MG10C13_044.png)

Kite

A kite is a quadrilateral with two pairs of adjacent sides equal.

A summary of the properties of a kite is:

  • Two pairs of adjacent sides are equal in length.
  • One pair of opposite angles are equal where the angles are between unequal sides.
  • One diagonal bisects the other diagonal and one diagonal bisects one pair of opposite angles.
  • Diagonals intersect at right-angles.
Figure 49: An example of a kite.
Figure 49 (MG10C13_045.png)

Other polygons

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

Table 8: Table of some polygons and their number of sides.
Sides Name
5 pentagon
6 hexagon
7 heptagon
8 octagon
10 decagon
15 pentadecagon
Figure 50: Examples of other polygons.
Figure 50 (MG10C13_046.png)

Extra

Angles of 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
(1)

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

Areas of Polygons

  1. Area of triangle: 12×12× base ×× perpendicular height
    Figure 51
    Figure 51 (MG10C13_047.png)
  2. Area of trapezium: 12×12× (sum of (parallel) sides) ×× perpendicular height
    Figure 52
    Figure 52 (MG10C13_048.png)
  3. Area of parallelogram and rhombus: base ×× perpendicular height
    Figure 53
    Figure 53 (MG10C13_049.png)
  4. Area of rectangle: length ×× breadth
    Figure 54
    Figure 54 (MG10C13_050.png)
  5. Area of square: length of side ×× length of side
    Figure 55
    Figure 55 (MG10C13_051.png)
  6. Area of circle: ππ x radius22
    Figure 56
    Figure 56 (MG10C13_052.png)

Figure 57
Khan Academy video on area and perimeter

Figure 58
Khan Academy video on area of a circle

Polygons
  1. For each case below, say whether the statement is true or false. For false statements, give a counter-example to prove it:
    1. All squares are rectangles
    2. All rectangles are squares
    3. All pentagons are similar
    4. All equilateral triangles are similar
    5. All pentagons are congruent
    6. All equilateral triangles are congruent
    Click here for the solution
  2. Find the areas of each of the given figures - remember area is measured in square units (cm22, m22, mm22).
    Figure 59
    Figure 59 (MG10C13_053.png)
    Click here for the solution

Exercises

  1. Find all the pairs of parallel lines in the following figures, giving reasons in each case.
    1. Figure 60
      Figure 60 (MG10C13_054.png)
    2. Figure 61
      Figure 61 (MG10C13_055.png)
    3. Figure 62
      Figure 62 (MG10C13_056.png)
    Click here for the solution
  2. Find angles aa, bb, cc and dd in each case, giving reasons.
    1. Figure 63
      Figure 63 (MG10C13_057.png)
    2. Figure 64
      Figure 64 (MG10C13_058.png)
    3. Figure 65
      Figure 65 (MG10C13_059.png)
    Click here for the solution
  3. Which of the following claims are true? Give a counter-example for those that are incorrect.
    1. All equilateral triangles are similar.
    2. All regular quadrilaterals are similar.
    3. In any ABCABC
       
      with ABC=90ABC=90 we have AB3+BC3=CA3AB3+BC3=CA3.
    4. All right-angled isosceles triangles with perimeter 10 cm are congruent.
    5. All rectangles with the same area are similar.
    Click here for the solution
  4. Say which of the following pairs of triangles are congruent with reasons.
    1. Figure 66
      Figure 66 (MG10C13_060.png)
    2. Figure 67
      Figure 67 (MG10C13_061.png)
    3. Figure 68
      Figure 68 (MG10C13_062.png)
    4. Figure 69
      Figure 69 (MG10C13_063.png)
    Click here for the solution
  5. For each pair of figures state whether they are similar or not. Give reasons.
    Figure 70
    Figure 70 (MG10C13_064.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 71
    Figure 71 (MG10C13_065.png)

    Click here for the solution

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