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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
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?
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