MATH325: Lecture 13

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Constructing a Congruent Triangle

Given a triangle ABC and a point D not on the triangle, construct a triangle DEF that is congruent to triangle ABC. Work in groups of 2 or 3 and write down the steps you use in your construction.

Hint: Lemma 3.3.3 allows you to construct circles such as CD(|AB|). Can you use this to construct edges for your triangle?

If you get stuck, feel free to ask another group for help.

Axiom 5, revisited

Lemma 3.3.4: Given a line AB and a point C, we can construct a line through C which is parallel to AB.

Discussion of Lemma 3.3.4: Wait a second! This is that impossibly complicated and important Axiom 5 that people spent so long trying to prove based on the first four axioms! If it's such a complicated axiom, why are we able to just go ahead and do this using a straightedge and compass?

The reason we can do this is that the surface we are working on is flat -- a white board or piece of paper. We don't have to prove that lines do or do not intersect; we can construct the lines and see whether they do or not. This construction would be impossible on the surface of a sphere.

The text book gives Lemma 3.3.3 as an exercise. How would you go about trying to prove it? Work on this construction in groups of 2 or 3.

We only have three points; what can we do with them?

The first thing we might think of is constructing the lines that join A and B to point C. If we had some way to copy angle BAC so that its vertex was at C, this might complete the proof. Maybe we can copy a triangle to C by copying each of its edge lengths (using our compass?)

Perhaps we decide we want to construct a line perpendicular to line AB, then a line perpendicular to that through C, resulting in a line through C parallel to line AB. We know how to construct perpendicular bisectors of segments (lemma 3.3.2). Can we use that?

We have three distances: |AB|, |BC| and |AC| which we could use to construct circles around the three points: A, B and C. Are any of these circles useful to us?