* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Angles An angle ∠AOB is the union of two noncollinear rays O r A
Survey
Document related concepts
Multilateration wikipedia , lookup
Integer triangle wikipedia , lookup
Cartesian coordinate system wikipedia , lookup
Duality (projective geometry) wikipedia , lookup
Riemann–Roch theorem wikipedia , lookup
Trigonometric functions wikipedia , lookup
Four color theorem wikipedia , lookup
Rational trigonometry wikipedia , lookup
Noether's theorem wikipedia , lookup
History of trigonometry wikipedia , lookup
Euler angles wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Transcript
Angles An angle r ∠AOBr is the union of two noncollinear rays O A and O B that share a common endpoint. In the same way that we can measure lengths by attaching numbers (distances) to line segments, we € can measure “rotational distance” spanned by € € angles. This is accomplished by means of angle axioms: [A-1] To every angle ∠AOB = ∠BOA we associate a real number strictly between 0 and 180 called the measure of the angle; this measure is labeled m ∠AOB = m∠BOA. € The use of the interval 0 < m∠AOB < 180 for angle measure is entirely a matter of convenience. We € could just as easily use the interval 0 < m∠AOB < π or even 0 < m∠AOB < 1. In particular, note the € absence of reference to degrees above; the intention here is to make use of the familiarity of degree € measure without needing to provide a special € definition for degree. The interior of ∠AOB is the set of all points in the plane of A,s O on the r and B that lie simultaneously sr A-side of OB and the B-side of OA , that is, sr sr € Interior ∠AOB = H(A, OB) ∩ H(B, OA ). € € € € sr Theorem If A-X-B and O does not lie on AB , then X is interior to ∠AOB. // r Corollary If A-X-B, then every point on O X , € except € for O, is interior to ∠AOB. // Recall that we used the additivity € of distances to define the betweenness of points (A-B-C means € AB + BC = AC); we would like to do the same for angles. For this, we need the Angle Addition Postulate: [A-2] If C is an interior point of ∠AOB, then m∠AOC + m∠COB = m∠AOB. In fact, the converse is also true: € € € € € € € Theorem If m∠AOC + m∠COB = m∠AOB, then r ray O C must pass through an interior point of r ∠AOB. Indeed, every point on O C is interior to ∠AOB. € // From this raxiom, define betweenness for r we can €r rays: if O A , O B and O C are r three rays sharing r a common endpoint, then O C lies O A and r r between r r O B , expressed by writing O A -O C -O B , if + m∠COB = m∠AOB. Betweenness for €m∠AOC € € rays behaves in€a manner entirely€similarly to betweenness for points. The connection between € € € these relations can be seen in the following Theorem Suppose r r A, rB, C lie on line l and O is not on l. Then O A -O C -O B if and only if A-C-B. // Furthermore, in the same way that we were able to formulate a Ruler Postulate that gives coordinates € € € to points on a line, we can formulate a Protractor Postulate that gives coordinates to rays within a half plane: r [A-3] Given any ray O A , there exists a one-to-one correspondence between thes set r r of all rays O X on oner side of the line OA , together with the ray€O A itself, and the interval of real numbers 0 ≤ θ < 180 (the number r θ is called ther coordinate of€the ray O X rand we write € this) sorthat O A [0], and €O X [θ ] to signify r whenever O B [θ ] and O C€[ φ ], then € m∠BOC =|θ − φ |. € € € € Of course, theorems relating € the € corresponding € € coordinates and betweenness of rays are true (and € their proofs handled similarly as well): Theorem Howeversone the rays r coordinatizes r r on oner side of therline OA (where O r A [0]), r ifr O B [θ ], O C [ φ ] and O D [ ω ], we have O B -O C -O D if and only if either θ < φ < ω or ω < φ < θ . // € € € € € € € € € € € € € Theorem [Angle Construction] If ∠ABC, ∠DEF are two angles for whichr m ∠ABC < m ∠DEF r r , then r there is a unique ray E G such that E D - E G - E F and m ∠ABC = m ∠GEF . // € € € € Corollary rEvery angle has a unique bisector. € € € € O C is the bisector of ∠AOB if € (The ray € m∠AOC = m∠COB .) // r r Rays O A and O B are called opposite rays if € € A-O-B, that is, the union of the two rays is a line. It is easy to show the € € € r € Theorem Every ray O A has a unique opposing r ray O B . // Two angles ∠AOC and ∠COB are rsaid to form a € linear pair if theyrshare a side O C and their r remaining sides O A and O B are opposite rays. Note Postulate makes no € that the Protractor € provision for assigning coordinates to all three of € these rays simultaneously since the coordinates of € € all rays must be less than 180. For this, we need another axiom. Calling any two angles whose measures sum to 180 supplementary angles allows us to state the Linear Pair Axiom: [A-4] A linear pair of angles is supplementary. An angle is right if its measure is 90, and any two angles whose measures sum to 90 are called complementary. An angle is acute if its measure is less than 90 and obtuse if its measure is greater than 90. Two distinct lines l and m are perpendicular if they contain sides of a right angle and we write l ⊥ m to denote this. (We can also extend the defintion of perpendicularity to segments and rays if the lines that contain them are perpendicular.) € Theorem Perpendicular lines make four right angles at their point of intersection. // r If angles ∠AOC and ∠COB share side O C and have no interior points in common, we call the angles adjacent. sr € € €s r Theorem If BD s rmeets s r AC at the point B between A and C, then BD ⊥ AC if and only if the adjacent angles ∠ABD and ∠DBC have equal measure. // € € Theorem There is a unique perpendicular to a € given line at any point on that line. // € Two angles form a vertical pair if the sides of one are the opposing rays of the sides of the other. Theorem Vertical angles have equal measure. // Moritz Pasch, one of the nineteenth century geometers who, like Hilbert, worked on developing the axiomatization of geometry, posed the following statement as an axiom and used it to prove the Plane Separation Postulate as a theorem. € A triangle ΔABC is the union of three segments A B , B C and C A , called sides of the triangle; the points A, B and C, called the vertices of the triangle, must be noncollinear. € € Theorem [Postulate of Pasch] Let A, B, C be noncollinear points in a plane. If l is any line in this plane that passes through a point D interior to side A B , then l must pass through some point E on the triangle not on A B ; that is, E satisfies exactly one of the following three properties: • E is an interior point on side C A ; € • E = C; or € • E is an interior point on side C B . // € Theorem If A-X-B, then X is an interior point of ∠AOC. // € € € Theorem [Crossbar r Theorem] If D is interior to ∠ABC, then ray B D must meet segment A C . // € €