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Points, lines and angles

Module by: Free High School Science Texts Project. E-mail the author

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

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