T
he t
w
o angles
cr
ea
ted are al
supplementary.
T
wo sets of v
ertica
l
angles—notic
e how one
set is exac
tl
y the same as
the other, so we only hav
e
TW
O dif
f
er
ent angles to
dea
l with.
and you’ll see tha
t
the Z pa
ttern
and
the
N pa
ttern
ar
e pr
etty much
the same thing.
vertical angle patterns
Parallel lines often come with
helpful angle shortcuts
When you have a line which meets or crosses a set of parallel
lines, all the sets of (opposite) vertical angles that are created are
the same.
TIRE
TRACKS
56º
?º
So this angle here should
be the same—56º.
110º
110º
110º
110º
70º
70º
70º
70º
w
ays
This means that you can find more missing angles without
doing any math at all—just by recognizing a few patterns. The
F Pattern, the Z Pattern, and the N Pattern are formed
when parallel lines cross or meet another line.
N pattern
70º
70º
110º
70º
70º
110º
Rota
te the page
Z pa
ttern
F pa
ttern
The F, Z, or N can be
hidden
Sometimes they’re upside
down, back-to-front, or both!
Look out for F, Z, and N
patterns around parallel
lines for a shortcut to
finding missing angles.
40 Chapter 1
finding missing angles
It’s the moment of truth.
Find the crucial mystery angles, x and y, and prove once and for
all whether Charlie is our shooter. If they’re equal he’s guilty. If
they’re not then the case is at another dead end.
BULLET
PATH
MICKY
BENNY
FOOTPRINTS
18º
CRIME SCENE SKETCH
CASE GEO180
(ANGLES AS MEASURED
BUT DRAWING NOT TO
SCALE)
72º
CHARLIE
FOOTPRINTS
28º
158º
TIRE
TRACKS
56º
18º
62º
50º
50º
130º
130º
32º
xº
yº
you are here 4 41
sharpen solution
It’s the moment of truth.
Find the crucial mystery angles, x and y, and prove once and for
all whether Charlie is our shooter. If they’re equal he’s guilty. If
they’re not then the case is at a dead end.
BULLET
PATH
MICKY
BENNY
FOOTPRINTS
18º
72º
CHARLIE
FOOTPRINTS
28º
158º
TIRE
TRACKS
56º
18º
62º
50º
50º
130º
130º
32º
xº
yº
CRIME SCENE SKETCH
CASE GEO180
(ANGLES AS MEASURED
BUT DRAWING NOT TO
SCALE)
If Charlie shot Micky, then we should find
that angles x and y are EQUAL, because
that’s how the bullet fired from his gun
should bounce.
42 Chapter 1
finding missing angles
x)
y)
56º
TIRE
TRACKS
The tire tracks are parallel.
So these sets of vertically
opposite angles are the same.
56º
56º
56º
56º
T
urn the book to make i
t
easi
er t
o spot the F pattern.
56º
50º
xº
x is in a triangle with the 56º we just found
and 50º, so
x = 180º - (56º + 50º) = 74º
74º
32º
y forms a straight line with x and 32º, so
y = 180º - (74º + 32º) = 74º
yº
x = 74º and y = 74º
The angles either side of where the bullet bounced are equal—it all
adds up.
This means that we have proved that Charlie shot Micky, even if it was
Benny he intended to hit!
Nic
e one—y
ou
sol
v
ed the cr
ime!
32º
yº
you are here 4 43
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