Angles in Parallel Lines
Prior Knowledge:
• Angles on straight lines.
• Angles in triangles.
Parallel lines are equidistant – they never touch, however far they are extended. Sometimes, it
helps to think of a set of train tracks.
When a line intersects two parallel lines, different types of angle are formed. You will need to be
able to remember each of these types of angle so that you can explain the relationships between
different angles around parallel lines.
The first two types of angles do not require parallel lines, but are frequently used in these types
of question:
Angle on a Straight Line
Angles on a straight line always add to 180°.
Vertically Opposite Angles
Vertically opposite angles are equal.
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Angles in Parallel Lines
You can prove this yourself. Look at the example below. Angles on a straight line add to 180°, then
b must equal 180 – 80 = 100°. Using angles on a straight line again, if b is 100° then c must equal
180 – 100 = 80°. This is the same as the angle opposite!
80°
b
c
Corresponding Angles
Corresponding angles are equal. Look at each example below, and imagine the upper parallel line
is moved down towards the lower parallel line. You can see that each angle around the intersection
on the top line is the same as the corresponding angle around the intersection of the bottom line.
These angles
are equal.
Corresponding angles form an F shape. Notice that the F shape can be upside down or back to front.
Alternate Angles
Alternate angles are equal.
These angles
are equal.
You can prove this yourself. Look at the example on the left. If the top alternate angle was 60°,
then that corresponds to the bottom-left angle at the lower intersection, so that is also 60°. That
bottom-left angle is opposite the lower alternate angle, so that must also equal 60°.
Alternate angles form a Z shape. Notice that the Z shape can be back to front.
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Angles in Parallel Lines
Co-interior/Supplementary/Allied Angles
Co-interior angles add up to 180°. These angles are sometimes also called supplementary or
allied angles.
These two angles
add up to 180°.
Again, you can prove this yourself. Say the upper co-interior angle is 100°. That corresponds to
the bottom-right angle at the lower intersection. The bottom-right angle at the lower intersection
is on a straight line with the lower co-interior angle, so they must add to 180°. 180 – 100 = 80°. This
means our co-interior angles add to 180°.
Example One
Find the value of the missing angle marked a. Give a reason for your answer.
70°
a
70°
a
This is an example of alternate angles, which are equal.
Therefore, the value of angle a is 70°.
The question also asks for a reason. Simply writing ‘alternate angle’ or ‘Z’, ‘F’, ‘C’ angle is not
sufficient. You should explain that alternate angles are equal.
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Angles in Parallel Lines
Example Two
Find the value of the missing angles marked b and c.
65°
c
b
65°
c
b
65°
c
b
Here, we can see an example of a corresponding angle.
Remember corresponding angles are equal therefore,
angle b is also 65°.
Finally, to find the value of c, we subtract 65° from 180°.
180 – 65 = 115
c = 115°
Example Three
Find the value of the missing angle d. Give a reason for your answer.
110°
d
In some questions, you will have to use more than one rule to find an answer. These questions
normally have more than one possible answer.
In this case, we know the angle opposite 110° is also 110°, because vertically opposite angles are
equal.
This opposite angle is a co-interior angle with d and we know that co-interior angles add to 180°.
180 – 110 = 70°, therefore d = 70°. Note that we gave reasons for each step of our answer.
See if you can spot any other ways of answering this question. There are at least 4 more ways
using two steps, and even more if you answer it in three steps.
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Angles in Parallel Lines
Your Turn
1.
a b
d c
e f
h g
a.
Which angle is corresponding to angle g?
b.
Which angle is alternate to angle f ?
c.
Which angle is corresponding to angle b?
d.
Which angle is supplementary to angle c?
e.
Which angle is alternate to angle c?
f.
Which angle is corresponding to angle h?
2. For each of the following questions, find the value of the missing angle.
Give a reason for you answer.
a.
b.
105°
110°
a
b
c.
d.
140°
63°
x
e.
y
f.
a
78°
94°
c
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Angles in Parallel Lines
g.
h.
d
x
57°
135°
3. Find the value of the missing angles, marked x and y. Give a reason for each stage of your
workings.
125°
y
x
115°
4. Find the value of the missing angles, marked a, b and c. Give a reason for each stage of
your working.
a c
b
32°
128°
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Angles in Parallel Lines
5. Find the value of the missing angle, marked x. Give a reason for each stage of your workings.
x
128°
Challenge
Prove that the opposite angles of a parallelogram are equal.
a
b
x y
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