Proof chart to explain how r is parallel to t

Question

Given: r is perpendicular to s and t is perpendicular to s

Prove: r is parallel to t

Answer 1

Let us assume that r and t are not parallel, but intersect in point O. We note the crossing point of r and s as R and the crossing point of t and s as T. Now we consider the triangle ORT. As given, the angle between r and s ∠R = 90°, also ∠T = 90°. The angle O is not known, but ∠O > 0. The sum of the angles in triangle ORT is

∠O + ∠R + ∠T = ∠O + 90° + 90° = 180° + ∠O > 180°.

However, we know that the sum of angles in any triangle is 180°. This contradiction shows that our assumption that r and t have an intersection point is wrong. Hence r and t do not intersect and are parallel.