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Head First 2D Geometry by Dawn Griffiths, Stray

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similarity and congruence
Use what you know to find what you don’t know
We said it in Chapter 1, and it still applies now. Work from what
you do know to find out what you don’t know. Like the angles
between the arrows.
This angle completes
And once you’ve found
it, you can just copy it
to the other angles it is
congruent with.
And you know this angle here
(it’s an equilateral triangle
because those three arrows are
congruent).
And you know these
two angles here…
a 360º “whole turn.”
And if you don’t have what you need, add it!
You can add parallel or perpendicular lines to your
you are here 4 83
sketch to break down the missing angles into parts you
have the tools to find.
Adding a line
here creates a
Z pattern.
These two lines are
parallel (because we
say so!)
Then we can use
what we know about
these angles…
…to find
these angles.
Ready to kick some
serious design butt?
find all the angles
There are 60 angles on the band’s logo design. Use the space on the right to start working on
the sketch and calculate them all. How many of each different angle are there?
Feeling overwhelmed?
Don’t panic! Everything
you need is in your
Chapter 1 toolbox.
84 Chapter 2
similarity and congruence
you are here 4 85
exercise solution
There are 60 angles on the band’s logo design. Calculate them all. How many of each different
angle are there?
Each arrow head is an equil
a
tera
l tr
iangle,
wi
th 3 equa
l angles: 180º / 3 = 60º.
At the c
enter of the design the darker
arrows meet. Sinc
e they ar
e a
ll the same
size, their sides f
orm another equil
a
teral
tr
iangle, so those are 60º as well.
60º
21 angles ar
e 60º.
The tick mark
s indica
te
tha
t a
ll the angles wi
th
one tick ar
e the same size.
21
down,
only
39 t
o
go!
There ar
e 21 angles tha
t are 60º, 24 angles that ar
e 90º, 9
angles tha
t are 120º, and 6 that are 150º.
Here’s how y
ou can find them a
ll:
86 Chapter 2
similarity and congruence
The bottom of each arrow is a square, so
those have four right angles—90º.
That’s 18 right angles.
d
To find the remaining angles, let’s use
similarity and just work on a chunk of
the design that is repeated.
Angle d makes a whole turn (360º) with
two right angles and the 60º angle we
already found, so:
d = 360º - (90º + 90º + 60º)
= 120º (there are 3 of these)
d
60º
39 down, onl
y 21 t
o go!
42 down, onl
y 18 t
o go…
you are here 4 87

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