Plane Geometry - Bruce E. Shapiro
Plane Geometry - Bruce E. Shapiro Plane Geometry - Bruce E. Shapiro
90 SECTION 17. THE CROSSBAR THEOREMTheorem 17.2 (MacLane’s Continuity Axiom a ) A point D is in theinterior of angle ∠BAC if and only if the ray −→ AD intersects the interior ofthe segment BC.a MacLane [MacLane, 1959] takes this result as an axiom and uses this name because itimplies Birkhoff’s [Birkhoff, 1932] continuity axiom (theorem 20.2).Figure 17.6: Illustration of MacLane’s Continuity Axiom (theorem 17.2.)Proof. (⇒) Suppose that D is in the interior of ∠BAC.−→crossbar theorem, AD intersects BC.Then by theSince the intersection point lies on a ray that is interior to ∠BAC, it doesnot lie on an endpoint of BC (definition of interior; else it would lie on one ofthe two rays that define the angle, not the intersection of their half-planes);hence the intersection point is interior to BC.(⇐) Suppose that −→ AD intersects the interior of BC. Call the point ofintersection E.−→ −→ −→ −→ −→Hence B ∗ E ∗ C. By theorem 16.8, AB ∗ AE ∗ AC. Hence AE = AD is inthe interior of angle ∠BAC. Hence D is in the interior of ∠BAC.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
Section 18Linear PairsDefinition 18.1 Two angles ∠BAD and ∠DAC are called a linear pair−→ −→if AB and AC are opposite rays.Definition 18.2 Two angles ∠BAD and ∠DAC are called supplementaryangles if ∠BAD + ∠DAC = 180.Figure 18.1: Angles angles ∠BAD and ∠DAC form a linear pairLemma 18.3 (Linear Pair Lemma) If C ∗ A ∗ B and D is in the interiorof ∠BAE then E is in the interior of ∠DAC.Proof. Since D is in the interior of ∠BAE, E and D are on the same sideof line ←→ AB by the definition of interior of an angle.Since ←→ AB = ←→ AC, this means that E and D are on the same side of line ←→ AC,i.e.,E ∈ H ←→ D, ACBy the crossbar theorem, ray −→ AD intersects segment BE (see figure 18.2).91
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- Page 84 and 85: 80 SECTION 16. ANGLESFigure 16.1: T
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Section 18Linear PairsDefinition 18.1 Two angles ∠BAD and ∠DAC are called a linear pair−→ −→if AB and AC are opposite rays.Definition 18.2 Two angles ∠BAD and ∠DAC are called supplementaryangles if ∠BAD + ∠DAC = 180.Figure 18.1: Angles angles ∠BAD and ∠DAC form a linear pairLemma 18.3 (Linear Pair Lemma) If C ∗ A ∗ B and D is in the interiorof ∠BAE then E is in the interior of ∠DAC.Proof. Since D is in the interior of ∠BAE, E and D are on the same sideof line ←→ AB by the definition of interior of an angle.Since ←→ AB = ←→ AC, this means that E and D are on the same side of line ←→ AC,i.e.,E ∈ H ←→ D, ACBy the crossbar theorem, ray −→ AD intersects segment BE (see figure 18.2).91
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