Download Angles, parallel lines and transversals

6-C
Angles, parallel lines and
transversals
KEY CONCEPTS
Parallel lines are two or more lines that are simple translations of each other.
The distance between parallel lines is the same across their entire lengths; they
never intersect each other. The train tracks in the picture are parallel to each
other; they only appear to intersect at the horizon.
Transversal
Parallel
lines
The term ‘transverse’ means ‘crossways’. A line that intersects with a pair of
parallel lines is called a transversal. The road crossing the two parallel lines
in the diagram represents a transversal.
When a transversal cuts a set of parallel lines, a number of angles are created.
The table below shows how pairs of angles are related.
Angle identification
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Relationship
Corresponding angles are
positioned on the same side of
the transversal and are either both
above or both below the parallel
lines; think of them as F-shaped.
∠a = ∠b
Co-interior angles are positioned
‘inside’ the parallel lines, on the
same side of the transversal; think
of them as C-shaped.
Angles are
supplementary:
∠a + ∠b = 180°.
Alternate angles are positioned
‘inside’ the parallel lines on
alternate sides of the transversal;
think of them as Z-shaped.
∠a = ∠b
Vertically opposite angles are
created when two lines intersect.
The angles opposite each other
are equal in size; think of them as
X-shaped.
∠a = ∠a
∠b = ∠b
Maths XPRESS 8
Example
a
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b
b
a
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b
b
a
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EXAMPLE 1
Use the diagram at right to answer the questions below.
a b
c d
a Which angle is vertically opposite c?
e f
g h
b Which angle is alternate to d?
c Which angle is corresponding to g?
d Which angle is co-interior to f ?
e Which of the above pairs are equal and which are supplementary?
WRITE
a Vertically opposite means that the angle is opposite the
intersection from c. They are X-shaped.
a
c d
e f
g h
b
b Alternate angles are in a Z shape.
a
c d
e f
g h
c is corresponding to g.
b
d Co-interior angles are in a C shape.
a
c d
e f
g h
e is alternate to d.
b
c Corresponding angles are in an F shape.
a
c d
e f
g h
b is vertically opposite c.
d is co-interior to f.
b
e 1 Colour all the angles equal to a in blue:
• d is vertically opposite a
• e is corresponding to a
• h is vertically opposite e.
2
a
c d
e f
g h
b
Colour all the angles equal to b in red:
• c is vertically opposite b
• f is corresponding to b
• h is vertically opposite f .
Angle relationships formed between parallel lines and a transversal can help
us evaluate the size of unknown angles by:
identifying the relationship between a known angle and an unknown angle
using your knowledge of which pair of angles is either equal or
supplementary.
Chapter 6
Geometry
205
EXAMPLE 2
Use the diagram to calculate the value of the
pronumerals. Provide a reason for each answer.
a 63o
b
81o
c
d e
f
74o
WRITE
1
Angles a and 63° are supplementary so they
sum to 180°. Write an equation and solve for a.
2
Angles b and 63° are alternate, and alternate
angles are equal.
Angles b, 81° and c are supplementary so they
sum to 180°. Write an equation and solve for c.
3
5
Angles c and d are alternate and so are equal.
The triangle contains angles c, e and 74°. The
sum of the interior angles in a triangle is 180°.
6
Angles f and 74° are alternate and so are equal.
4
a + 63° = 180°
a = 180° − 63°
a = 117°
b = 63°
(supplementary)
(alternate)
63° + 81° + c = 180° (supplementary)
144° + c = 180°
c = 180° − 144°
c = 36°
d = c = 36°
(alternate)
c + e + 74° = 180° (interior angles of a
triangle)
36° + e + 74° = 180°
110° + e = 180°
e = 180° − 110°
e = 70°
f = 74°
(alternate)
LEARNING EXPERIENCE
‘Identify the angle’ race
Equipment: string, scissors, Blu Tack, stopwatch, paper, pen
1 Set up the classroom with four rows of tables.
2 Form groups of four students and collect three pieces of string and a blob of Blu Tack for each person.
3 Each group lines up along one of the rows, with one person per table. Each person uses the string and
Blu Tack to model one of the four types of angles represented in Key concepts. Use blobs of Blu Tack
to show the positions of the angles. (Use a small amount of Blu Tack to hold the string in place.)
4 Remove one of the blobs of Blu Tack showing the position of the angles. A competitor from another
group completes the problem by putting the blob of Blu Tack back in the appropriate position. Group
members who have modelled an angle stand behind their angle during the race.
5 Now the race begins! A group of four lines up, with one group member at the top of each row.
When the timekeeper says ‘Start’, timing starts and does not finish until the last group member has
completed all their problems. When the competitor reaches a model, the student who constructed the
model tells the competitor the kind of angle they need to make with the second blob of Blu Tack.
6 Groups take turns to race through the tables and make the models. Points are based on speed and
accuracy. Each second is worth 1 point and every incorrect answer is worth an additional 5 points.
The group with the fewest points wins.
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Maths XPRESS 8