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exercise solution
a
b
c
b
The dark triangle is turned 60º compared to the
light one, and each angle of the triangle is 60º so
these edges are parallel. A line which is perpendicular
to one line is also perpendicular to lines parallel to
that line, so b is also 90º (another 6).
a
c
To find angle c, draw a
line parallel to the base of
the dark triangle.
60º
This creates some equal angles—the Z pattern. Angle c is
this Z angle (60º) plus the perpendicular (it’s one side of
the square and we drew our line parallel to the bottom of
the square), so
c = 60º + 90º = 150º (6 of these, too!)
48 down, 12 t
o go....
54 down…jus
t six left…
a
150º
To find angle a, first find the angles of this triangle.
This angle is the supplement of a right angle we already found, so it’s 90º.
This angle is the supplement of the 150ºwe just found, so it’s 30º.
The angle next to a is the remaining angle in the triangle = 60º, so a is the
supplement of this angle = 180º- 60º = 120º (6 of these as well).
…continued
88 Chapter 2
similarity and congruence
Tonight’s talk: Similarity and Congruence
Similarity:
I’m worried about you. You seem neurotic, always
concerned about being a size zero, or seven, or
whatever. You need to relax more.
I just think if you stopped worrying so much about
size and focused on proportionality you’d have more
opportunities. I get used all the time—my flexibility
is a real asset.
Faster? How? I’m pretty fast, you know!
True. And I’m a big stickler for efficiency. People
would do well to use both of us more!
Oh, tell me about it! I think they forget that we’re
not all about the triangles. And that we travel so
well.…
Congruence:
But size DOES matter. It’s all very well having your
angles right on paper, but if something is too big or
too small, in the real world size actually matters.
And potentially a real headache! Sometimes it
doesn’t cut it just being similar—and that’s where
I come in, when you need to rely on size as well as
shape. Plus, I’m faster to use.
Yeah, you’re quick, but even if something’s similar,
if it’s not congruent, then you’ve gotta do some
math for the lengths. OK, it’s only multiplication
and division, but it all takes time.
Yes. Are you getting much non-triangle work these
days? I’m mostly getting triangle stuff still, and it’s
not that I don’t like it…but you know, I have so
much more potential.
Completely—I love to travel! You can flip me
upside down, back to front, spin me around and
move me from one place to another, and I still work
just as well.
Great to catch up! I’ll let you know if I get any work
you could help out with.
you are here 4 89
expanding your design
Hey—I had a cool idea. Is there any
way you could work up the design so
that maybe we could get some t-shirts
printed, too? The band would be psyched!
All the arr
ows ar
e the same.
The squ
are par
t is ha
lf the
length of the arr
ow head par
t.
T
ria
ngle sides ar
e the same.
The darker set of arr
ows ar
e
3/4 size.
It’s an interesting suggestion—could you represent the
design in a way that means that you could use it to etch
gadgets of any size, and maybe even print it on T-shirts?
Can y
ou r
eally get
this diagram t
o fi
t
p
er
f
ec
tl
y onto a
ll of
these wi
thout doing a
t
on of w
ork f
or each
dif
f
er
ent size?
90 Chapter 2
similarity and congruence
It would be cool to finish marking up your design of the band’s logo so that it would
fit on any size gadget—even a T-shirt.
But what does that actually mean you need to do?
a) Draw the design at different sizes and mark up different lengths on each of
them, that way you’ll always have one to fit.
b) Forget the final size, and just make sure you’ve captured the relationships
between the lengths of the lines, so it stays proportional.
Wr
i
te what y
ou’d
need t
o do and
why you do or
don’t think i
ts
the
bes
t approach.
I think you’re getting
a bit ahead of yourself
here—we don’t have any lengths
at all yet. It’s way too early to be
thinking about resizing it…
you are here 4 91

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