s

traight line

angles

Equal angles

T

hese ar

en’t

the angles y

ou

f

ound bef

or

e.

finding missing angles

Use what you know to

find what you don’t know

By spotting patterns around the angles that you do know, you can

work out the angles that you don’t yet know. Sometimes, like with

vertical angles, you don’t even have to do any math.

Here are some patterns for you to look out for:

Complementary

Angles on a

T

r

iangles

V

ertica

l angles

On the sketch, mark all the new angles you need to find to solve

the crime (there are at least 11).

BULLET

PATH

MICKY

BENNY

FOOTPRINTS

BROKEN

GLASS

18º

CRIME SCENE SKETCH

CASE GEO180

(ANGLES AS MEASURED

BUT DRAWING NOT TO

SCALE)

72º

CHARLIE

FOOTPRINTS

28º

158º

TIRE

TRACKS

56º

W

e’ll assume tha

t this is a

s

traight line and w

ork back

t

ow

ard the bounc

e angles.

you are here 4 31

moving beyond triangles

Mark on the sketch all the new angles you need to find to solve

the crime (there are at least 11).

BULLET

PATH

MICKY

BENNY

FOOTPRINTS

BROKEN

GLASS

18º

CRIME SCENE SKETCH

CASE GEO180

(ANGLES AS MEASURED

BUT DRAWING NOT TO

SCALE)

72º

CHARLIE

FOOTPRINTS

28º

158º

TIRE

TRACKS

56º

1

2

3

4

5

67

8

9

10

11

OK, angles 1 and 2,

we can find, but what

about angle 3?

Jill: What’s the problem? We just keep going like we have been.

Frank: But the room isn’t triangular—it has four sides.

32 Chapter 1

158º

28º

Charlie

The room

has four

sides.

This shape

also has

four sides.

finding missing angles

Jill: Oh, good point. Could we add a line of our own and chop the room into two

triangles like this?

158º

Joe: How on earth does that help?

Jill: Well…at least we know that angles in a triangle add up to 180 degrees. So if

we make triangles out of the room, we can keep going like we have been. In fact,

there’s another shape with four sides there that could help us with angle 3—we

could divide that into triangles as well maybe? Like this?

158º

Joe: OK, but that seems like a lot more work. Maybe we could try that

thing we did with the paper triangle? With the corners? That might show

us something about four-sided shapes in general, and we can use that to

find those missing angles?

What is it that you need to find out about four-sided shapes in

order to find the missing angles?

See whether you can figure it out, either by adding up the

angles using two triangles, or tearing the corners from a four-

sided shape to see what they add up to.

you are here 4 33

sharpen solution

What is it that you need to find out about four-sided shapes in

order to find the missing angles?

See whether either method can help you figure it out.

To find the missing angles, we need to know

what angles in a four-sided shape add up to.

A

B

C

D

E

F

One way of finding out is to split the four-

sided shape into two triangles.

A + B + C = 180º

D + E + F = 180º

A + B + C + D + E + F = 180º + 180º

= 360º

Using a different method, if you cut out a

four-sided shape and tear the corners off,

and put them together with the points in the

middle, you’ll find they make a whole turn.

And a whole turn

(two half turns) is 360º.

Both methods give us the

same answer: 360º.

34 Chapter 1

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