Plane Geometry - Bruce E. Shapiro

SECTION 30. TRIANGLES IN NEUTRAL GEOMETRY 139Theorem 30.4 Let △ABC be a triangle. Then there exists a point D ∉←→AB such thatσ(△ABD) = σ(△ABC)and the angle measure of one of the interior angles of △ABD is less thanor equal to 1 2 µ(∠CAB).Proof. See figure 30.3.Let E be the midpoint of BC (existence of midpoints, theorem 14.29).By the ruler postulate we can define a point D ∈ −→ AE such that A ∗ E ∗ Dand AE = ED.By the vertical angle theorem (theorem 19.7), δ = γ, hence by SAS,△AEC ∼ = △DEB. Henceσ(△AEC) = σ(△DEB) (30.1)because the angle sums of congruent triangles are equal. By theorem 30.3,Figure 30.3: Proof of Theorem 30.4.σ(△ABE) + σ(△AEC) = σ(△ABC) + 180σ(△ABE) + σ(△BED) = σ(△ABD) + 180Subtracting the second equation from the first,σ(△AEC) − σ(△BED) =σ(△ABC) − σ(△ABD) (30.2)Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.