### MATH325: Lecture 13

**Syllabus | Schedule | Grading Keys/Rubrics
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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 C_{D}(|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?