Head First 2D Geometry

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

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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

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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…

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