Undefined Terms Birkoff’s Postulates Some Defined Terms Euclidean Parallel Postulate
point line (set of points) distance (a map from pairs of points to R≥0 ) angle (a set of points defined by an ordered pair of rays with a common vertex)
Existence of Perpendiculars Parallel Postulate Becomes a Theorem
Birkoff’s Postulates Postulate I: Postulate of Line Measure. A set of points fA, B...g on any line can be put into a 1:1 correspondence with the real numbers fa, b...g so that jb − aj = d(A, B) for all points A and B. Postulate II: Point-Line Postulate. There is one and only one line, `, that contains any two given distinct points P and Q. Postulate III: Postulate of Angle Measure. A set of rays fl, m, n...g through any point O can be put into 1:1 correspondence with the real numbers a(mod2Π) so that if A and B are points (not equal to O) of l and m, respectively, the difference am − al (mod2Π) of the numbers associated with the lines l and m is the measure of ∠AOB. Postulate IV: Postulate of Similarity. Given two triangles ABC and A0 B 0 C 0 and some constant k > 0, d(A0 , B 0 ) = kd(A, B), d(A0 , C 0 ) = kd(A, C ) and ∠B 0 A0 C 0 = ∠BAC , then d(B 0 , C 0 ) = kd(B, C ), ∠C 0 B 0 A0 = ∠CBA, and ∠A0 C 0 B 0 = ∠ACB
Birkoff’s Postulates Some Defined Terms Euclidean Parallel Postulate Existence of Perpendiculars Parallel Postulate Becomes a Theorem
Defining Between and Ray Birkoff’s Postulates Some Defined Terms
Between Let A, B, C be points. Then B is between A and C if d(AC ) = d(AB) + d(BC ). Ray ~ is the set of Let O and A be two points. Then the ray OA all points B such that either A is in between O and B or B is between O and A.
Euclidean Parallel Postulate Existence of Perpendiculars Parallel Postulate Becomes a Theorem
Euclidean Postulate If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
Birkoff’s Postulates Some Defined Terms Euclidean Parallel Postulate Existence of Perpendiculars Parallel Postulate Becomes a Theorem
↔
↔
Line BD and line AC will intersect if m∠CAD + m∠ADB < 180◦
Existence of a Perpendicular Birkoff’s Postulates Some Defined Terms Euclidean Parallel Postulate Existence of Perpendiculars Parallel Postulate Becomes a Theorem
↔
↔
We will assume that given a line BC and a point A 6∈ BC there exists a line passing through A and perpendicular to ↔
BC .
No need for parallel postulate Birkoff’s Postulates Some Defined Terms Euclidean Parallel Postulate Existence of Perpendiculars Parallel Postulate Becomes a Theorem
Given a point A ↔
↔
a line DF with A not on DF , ↔ ↔ DF ⊥ AD ↔ AB m∠DAB < 90◦ ↔
↔
We need to show that DF and AB intersect.
No need for parallel postulate Birkoff’s Postulates Some Defined Terms Euclidean Parallel Postulate Existence of Perpendiculars Parallel Postulate Becomes a Theorem
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3
Drop a perpendicular from B to C . jDE j Find a point E on DF so that jADj jAC j = jBC j . This will make triangles 4ADE and 4ABC similar. Thus m∠EAD = m∠BAC and protractor postulate implies that ↔
↔
AE and AB are the same.
