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Name: ___________________ Straight and Symmetries Dialogue 3 Page 1 Reactions 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- Name: ___________________ Straight and Symmetries Dialogue 3 Page 2 Reactions 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 Name: ___________________ Straight and Symmetries Dialogue 3 Page 3 Reactions 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? Name: ___________________ Straight and Symmetries Dialogue 3 Mathematician: Page 4 Reactions 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? Name: ___________________ Straight and Symmetries Dialogue 3 Page 5 Reactions 6. When two lines cross, does the size of the angles they make a difference? Why or Why not?