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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