Existence of Right Angle in Hilbert Axioms

Filipinacws

Filipinacws

Answered question

2022-08-12

Existence of Right Angle in Hilbert Axioms
Hilbert, in Foundations of Geometry briefly mentions that the existence of right angles is a corollary to the supplementary angle theorem. (i.e. If two angles are congruent, then their supplementary angles are congruent).
How does existence of right angles follow from this?

Answer & Explanation

Ezequiel Davidson

Ezequiel Davidson

Beginner2022-08-13Added 11 answers

Step 1
It's not an immediate conclusion from this fact. However, this fact is used in the proof. The proof goes as follows:
Take a line L and a point p not lying on L. Next take a point a L and choose a ray A with origin a which is contained in L. Let P := a p and let M be a halfplane with boundary L, to which point p belongs. Next lay off a ray Q with origin a contained in the halplane M (i.e. halfplane complementary to M) such that A Q A P. Next take a point q Q such that a q a p. Since p M and q M , the segment p q ¯ and line L have a point in common. Call it c. Now we have three cases:
Step 2
1. c = a. Then angle AP is supplementary to AQ and they are congruent. Hence they are right angles.2. c A. Then by SAS c a p c a q. From this congruence we get a c p a c q and these angles are supplementary. Hence they are right angles.
3. c A . Here we use the aforementioned fact. A P , A P and A Q , A Q are pairs of supplementary angles and A P A Q. So c a q = A Q A P = c a p. Again by SAS c a p c a q and the rest is the same as in case 2.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in High school geometry

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?