Y

ou can tell these ar

e the

same angle, ev

en bef

ore y

ou

work out wha

t tha

t angle is.

spotting similar triangles

Circle the triangles below that you can be SURE are similar to the triangle repeated

in the design.

53

o

86

o

53

o

41

o

86

o

53

o

53

o

41

o

41

o

53

o

86

o

86

o

41

o

86

o

41

o

You can spot similar triangles based on

just two angles

We know that angles in a triangle add up to 180 degrees, so once

you’ve got two angles in each triangle, you can always work out the

third.

And if you’ve noticed that two angles in one triangle are equal to two

angles in another triangle, then you can tell the triangles are similar

without even doing any math!

41

o

86

o

Look for two

equal angles

to spot similar

triangles.

41

o

86

o

62 Chapter 2

similarity and congruence

Q:

What if the triangles are flipped, so one has a 41° on the

right and the other has a 41° angle on left? Are they similar?

A:As long as you can spot another angle which is in

both triangles then yes, they’re definitely similar. Similarity is

maintained even if your shape is reflected or rotated.

Q:

Isn’t using similarity kind of like cheating? Shouldn’t I

be working out all the angles individually?

A:Cheating? We like to think of it as working smarter rather

than harder. It does save you plenty of leg work though. Most

geometry teachers will be more impressed by use of similarity

than repetitive calculations anyway—just make sure to make a

note on your work that you used similarity.

How many repeated angles are there in total on this diagram,

including the ones you’ve already marked plus the angles a

through i? (Count each value once—if it’s repeated don’t count it

again.)

41º 53º

86º

86º

41º

41º

41º

53º

53º

a

c

e

d

i

f

g

b

This corner a

lso

90

b

you are here 4 63

sharpen solution

a) The diagram tells us angle a is 90

o.

b) angle b completes the four-sided shape, so it must be

360

o

- (90

o

+ 90

o

+ 90

o)

=

90

o

c) angle c is on a straight line with 41

º

, so C = 180º - 41º = 139º

d) angle d is also on a straight line with 41º, so d = 180º - 41º = 139º

e) angle e is on a straight line with 53º, so e = 180º - 53º = 127º

90

o

b

41º

41º

53º

C

d

e

How many repeated angles are there in total on this diagram,

including the ones you’ve already marked, plus the angles a

through i? (Count each value only once—if it’s repeated don’t count

it again.)

Angles in a four-sided s

hap

e

a

lw

ays add up t

o 360º.

41º 53º

86º

86º

41º

41º

41º

53º

53º

a

c

e

d

i

f

g

b

This corner a

lso

90

b

64 Chapter 2

A bunch are r

ep

eated, r

ight? W

er

e

y

ou sur

pr

ised?

Z s

hap

e bet

w

een

para

llel lines

because they ar

e a

t the same

angle: 41º.

T

hese two s

hap

es

sets of matching angles.

similarity and congruence

Different angles: 90

º

, 41

º

, 53

º

, 86

º

, 127

º

, 139

º

, 94

º

… = 7 different angles in total

Repeated angles: 90

º

, 41

º

, 53

º

, 86

º

, 127

º

, 139

º

… = 6 angles are repeated

f) The left sides of the two mountains are both at 41

º

,

so they must be parallel. This means that angle f makes a

Z shape (alternate angles) with the 86

º

peak, so f must

also be 86

º

.

86º

f

T

hese two lines are parallel

g) by similarity, angle g must be the same as angle C, so g = 139

º

ar

e simil

ar, so their

angles ar

e the same!

T

hose li

ttle tick mark

s show

h) by similarity, angle i must be the same as angle e, so i = 127

º

i) angle i is on a straight line with f, which is 86

º

, so h = 180

º

- 86

º

= 94

º

you are here 4 65

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