Question

4.2.11 Prove Theorem 4.2.9: If corresponding angles of omega triangles are congruent, then the omega triangles are congruent.

Answer 1

Assume that in omega triangle △ABΩ△ABΩ we have ∠A≅∠B∠A≅∠B.

Let MM be the midpoint of segment AB¯¯¯¯¯AB¯, let ll be a line perpendicular to AB←→AB↔ passing through MM, and let rr be a ray emanating from MM contained in ll which lies on the same side of AB←→AB↔ as rays AΩ−→−AΩ→ and BΩ−→−BΩ→.

We shall prove that rays rr and AΩ−→−AΩ→ are limiting parallel (by the same argument rr and BΩ−→−BΩ→ will be limiting parallel).

In the first step suppose that rays rr and AΩ−→−AΩ→ intersect at point PP. Then triangles △AMP△AMP and △BMP△BMP are congruent and therefore ∠ABΩ=∠B≅∠A=∠BAP=∠MAP≅MBP=∠ABP∠ABΩ=∠B≅∠A=∠BAP=∠MAP≅MBP=∠ABP and by one of the axioms rays MP−→−MP→ and MΩ−→−MΩ→ are equal. Hence PP is a point common to rays AQ−→−AQ→ and BQ−→−BQ→ which contradicts the assumption that these rays are parallel. We conclude that rays rr and AΩ−→−AΩ→ are disjoint.

Hense the theorem 4.2.9