Question

In class, we saw that Plane Geometry + EPP (as an axiom) implies the Pythagorean theorem (as a theorem). In this question, complete the following steps to prove that Plane Geometry + Pythagorean Theorem (as an axiom) implies the Euclidean Parallel Postulate (as a theorem), showing that the two statements are equivalent in Plane Geometry. The following parts should be proved in Plane Geometry + Pythagorean Theorem. Remember to not assume EPP or its consequences, as that's what you're trying to prove! (a) Show how to use the axioms of Plane Geometry to explicitly construct a triangle △ABC with right angle at C and ∣ AC ∣=∣ BC ∣=1. (b) Drop an altitude from C to intersect AB at D and show that CD≅AD≅BD and that ∠CBA≅ ∠BCD≅∠ACD≅∠CAB (c) Explain why part (b) implies that the measures of the angles of △ABC sum to 180∘

. (d) Use the fact that you've shown that a triangle exists whose angle sum is 180∘ to argue that the Euclidean Paralle Postulate must hold.

Answer 1

Using the Pythagorean Theorem and the axioms of Plane Geometry, one can construct a right-angled isosceles triangle, demonstrate the equality of its altitude to its legs, conclude that the angles sum to 180 degrees, and generalize to prove the Euclidean Parallel Postulate.

Step-by-step explanation:

Answering the question of how Plane Geometry plus the Pythagorean Theorem implies the Euclidean Parallel Postulate (EPP):

1. First, construct triangle △ABC with a right angle at C using Plane Geometry axioms. Make AC and BC each equal to 1 unit.
2. Next, drop an altitude from C to AB, meeting it at point D. With the Pythagorean Theorem a² + b² = c², and both a and b equal to 1, the lengths of AD and BD will each equal 1, as the hypotenuse of these right triangles (AC and BC) is equal to the length of the sides (AD and BD). Hence, CD ≅ AD ≅ BD. Moreover, all acute angles in these triangles will be 45°, making ∠CBA ≅ ∠BCD ≅ ∠ACD ≅ ∠CAB.
3. This construction shows that the sum of the angles in △ABC is 180°, since ∠ACB is 90° and ∠CAB and ∠ABC are 45° each. The angle sum property is consistent with Plane Geometry.
4. Finally, by generalization, if one triangle exists with an angle sum of 180°, then this must hold for all triangles in Plane Geometry. Hence, the EPP is implied, which states that parallel lines do not converge and thus the sum of angles in any triangle is 180°.