Rational numbers and similarity

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In this activity we play a game of “what if” and see a reason that the ancient Greeks might have wanted every number to be rational.

Think about plain old plane geometry. What are some theorems that you would want to be true?

What are the basic theorems involving similar triangles?

OK—now we are going to do something very strange. Let’s suppose that every number is rational. In essence, let’s put ourselves into the mindset of the ancient Greeks, before they knew that irrational numbers existed.

Suppose that you have two triangles whose angles are congruent. Can you make a fairly simple argument, using the fact that the sides are rational numbers, that shows that the sides are proportional? Hint: You may need to use ASA.

Suppose that you have two triangles whose sides are proportional. Can you make a fairly simple argument, using the fact that the sides are rational numbers, that shows that the angles are congruent? Hint: You may need to use SSS.