Plane Geometry - Bruce E. Shapiro

SECTION 24. EXTERIOR ANGLES 117Theorem 24.5 (Uniqueness of Perpendiculars) For every line l andfor every point P ∉ l, there exists exactly one line m ⊥ l such that P ∈ m.Proof. See figure 24.3. Suppose P is a point with P ∉ l and let m ⊥ lthrough P .Suppose n ⊥ l is a second line, distinct from m, through P . (RAA assumption)Let R be the point at which n intersects l, Then R ≠ Q. (Otherwise,R ∈ m and there would be two distinct points, R and P , at which m andn intersect. This would mean that m = n which would violate the RAAassumption.)Triangle △P RQ has an exterior angle β = 90 and an interior angle α = 90at R by the RAA assumption. This contradicts the exterior angle theorem(theorem 24.4) which says that β > α. Hence we reject the RAA hypothesis.Figure 24.3: Since there can be only one perpendicular line to l throughthe point P , β ≠ 90.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.