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Straight and Symmetries Dialogue 3
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Mathematician: When two lines intersect how many
angles less than 180o are formed?
c
a
b
d
Student 1:
a, b, c, d
There are four angles – the ones labeled
in the picture.
Mathematician:
same?
Good.
Are any of these angles the
Student 2: Well, it looks like angles a
are the same size and also angles c and
and
d.
b
Mathematician: How could you show that the angles
and b have the same size?
Student 3: We could see if angle
to fit exactly on angle b.
a
a
can be placed
Student 1: I remember that we said that if two
figures could be placed to fit exactly on top of each
other then we called them “congruent”.
Mathematician:
congruent?
So, how can we see if they are
Student 2: We could make a copy of angle
if it fits exactly on b.
Student 3:
.
a
and see
We have already made a copy of angle a.
Student 1: In our activity we showed that the figure
formed by the two lines has ½-turn symmetry. We
should be able to use that.
Student 2:
We know that straight lines have half-
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Straight and Symmetries Dialogue 3
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turn symmetry about every point and thus both lines
have half-turn symmetry about the intersection point.
So take the region of the plane that neighbors the
intersection point and rotate it through a half-turn.
Then both lines will rotate onto themselves and thus
angle a will rotate to angle b since they are
regions between the lines.
Student 3:
The same happens with
Mathematician:
Angle Theorem.
c
and
d.
Mathematicians call this the Vertical
Vertical Angle Theorem. If two lines intersect then the opposite
angles are the same size.
Mathematician: The argument that Student 2 gave is
called a “proof” of this theorem. It is convincing
and it explains why the statement is true.
Student 1: We also saw in the activity that the
figure of the two lines has mirror symmetry about a
line that is in the middle of angle c.
Student 2: Yes, and we found that line by folding
the tracing paper so that the two sides of angle c
lay on top of each other.
Mathematician: Good. This fold line is called the
angle bisector of angle c.
Student 3: And if we fold on this line then angle
seems to lie exactly on top of angle b.
a
Student 1. I am a little confused. When we fold
over so that the two sides of the angle c are on
top of each other, does this mean that the two sides
of angle d have to lie on top of each other?
Mathematician: Good question. To talk about it,
let’s label the picture differently:
A
C
O
Student 1: OK, with this picture my question is: If
we fold across the bisector of the angle AOC, then
B
D
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Straight and Symmetries Dialogue 3
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the segment OC will lie on top of the segment
But why will the segment OD lie on top of OB?
OA.
Mathematician: Excellent question. To answer the
question we need to think of reflecting instead of
folding. Put the labels A, B, C, D, O on your traced
figure and original figure. Now flip over the tracing
paper so that the traced OC lie on top of the
original OA and the traced OA lies on top of the
original OC. If now OD does not lie on top of OB then
we would have something like the following situation
where OD is not on OB:
C = traced A
O
traced D
B
Student 3: But this is not possible, because if COB
is not turning then the traced DOA would be turning.
Student 2: Also, in this picture it is not possible
for both COB and the traced DOA to have mirror
symmetry in the line because with mirror symmetry
both sides of the line must be the same.
Mathematician: Very good. This is a second proof of
the Vertical Angle Theorem.
Student 1: Why did we draw the picture with the
lines not perpendicular? Won’t the proofs work for
perpendicular lines also?
Student 2: When the lines are perpendicular then it
seems that all four angles are equal.
Mathematician: In fact this is the original
definition by Euclid over 2 thousand years ago: Two
line are perpendicular if the four angles formed are
all congruent.
Student 3: Are there different symmetries for
perpendicular lines?
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Straight and Symmetries Dialogue 3
Mathematician:
Page 4
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Let’s find out.
1. What symmetries did the figure not have?
How did you know this?
2. At the top of page two where student 2 says “We know straight lines
have…” Does this explanation of why the Vertical Angle Theorem
works convince you? Why?
3. Show what is happening in Student 2’s explanation with a picture.
4. What is an angle bisector?
5. In your own words and pictures, how does reflection symmetry show
that the Vertical Angle Theorem also works?
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Straight and Symmetries Dialogue 3
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6. When two lines cross, does the size of the angles they make a
difference? Why or Why not?