Axiom A-1: To every angle there corresponds a unique, real number, 0 < < 180.

Transcription

1 Axiom A-1: To every angle there corresponds a unique, real number, 0 < < 180. We denote the measure of ABC by m ABC. (Temporary Definition): A point D lies in the interior of ABC iff there exists a segment with E-D-F, and such that neither E nor F are the point B, and. Axiom A-2: (Angle Addition Postulate) If D lies in the interior of angle ABC, then m ABC = m ABD + m DBC The book takes as part of this axiom the converse of this statement, but we can actually prove it after the next section.

2 Definition: Suppose that,,and are concurrent rays, all having the same endpoint O. Then ray is said to be between rays and, written - - if and only if these rays are distinct, and if m AOB + m BOC = m AOC, Note: This tells you that the ability to add angle measures is equivalent to betweenness for rays. In practice, what this says is that before you add angle measures, be sure the appropriate betweenness relationships are in place.

3 Axiom A-3: Protractor Postulate. The set of rays having a common origin O and lying on one side of line l containing ray, may be assigned to the real numbers for which 0 < 180, called coordinates, in such a manner that: (1) each ray is assigned a unique coordinate. (2) each coordinate is assigned to a unique ray. (3) the coordinate of is 0. (4) if rays and have coordinates and, then m POQ = -. Note: This connects angle measures to real numbers, so there are now as many rays coming off a point in one half plane as there are real numbers between 0 and 180.

4 Using the protractor postulate, we are able to create angles with various properties. In particular, we can: Copy angles. Bisect angles. Double angles. We prove a particular theorem of this kind below, using a little lemma first: Lemma: Let rays have coordinates a, b, and c respectively. Then iff a < b < c or c < b < a. Proof: Just like Theorem 3 in section 2.4.

5 Theorem 1. Angle Construction Theorem: Given any two angles ABC and DEF such that m ABC < m DEF, there is a unique ray such that m ABC = m GEF and. Sketch of proof: Let a = m ABC and b = m DEF, so 0 < a < b < 180. Using the protractor postulate, assign coordinates to rays on the D-side of with 0 assigned to. The coordinate of must then be b. Make the unique ray with coordinate a. By lemma above,. Then m GEF is a-0 = a = m ABC.

6 Basic Facts About Angles: Using the ruler postulate, it is easy to show that If A-B-C, is the line containing A, B, and C. We call such rays opposing or opposite rays. We note that given any ray, we can always find a unique opposing ray. (How?) Definition: If the sides of one angle are opposite rays to the respective sides of another angle, the angles are said to form a vertical pair. Definition: Two angles are said to form a linear pair iff they have one side in common and the other two sides are opposite rays. We call any two angles whose angle measures sum to 180 a supplementary pair, or more simply, supplementary, and two angles whose angle measures sum to 90 a complementary pair, or complementary.

7 Theorem 2: Two angles which are supplementary (or complementary) to the same angle have equal angle measures. Outline of proof: Suppose angles and are both supplementary to angle. Then m + m = 180 = m + m. Canceling m from both sides gives the result. Proof for complementary case is similar. Now another Axiom that we need to make our geometry work: Axiom A-4. A linear pair of angles is a supplementary pair.

8 Definition: An angle having measure 90 is called a right angle. Angles having measure less than 90 are called acute angles, and those with measure greater than 90, obtuse angles. Definition: If line l intersects another line m at some point A and contain the sides of a right angle, then l is said to be perpendicular to m, and we write l m. Lemma: One line is perpendicular to another line iff the two lines form four right angles at their point of intersection. Outline of proof: Use the linear pair axiom. Corollary: The relation of perpendicularity is symmetric, that is, if l m, then m l.

9 Theorem 3: If line meets segment at an interior point B on that segment, then iff the adjacent angles at B have equal measures. Proof: Easy. Theorem 4 (Existence and Uniqueness of Perpendiculars): Suppose that in some plane line m is given and an arbitrary point A on line m is located. Then there exists a unique line l that is perpendicular to m at A. Outline of proof: Locate a point B A on m. Use the protractor postulate to establish as having coordinate 0. Find the unique ray having coordinate 90. Then line and line m contain the sides of a right angle and are perpendicular by definition. Uniqueness follows from the protractor postulate.

10 Theorem 5 (Vertical Pair Theorem) Vertical angles have equal measures. Outline of proof: Referring to the diagram, angles 1 and 3 form a linear pair, as do angles 2 and 3. By axiom A-4, angles 1 and 2 are each supplementary to a common angle 3, and by Theorem 2, must have equal measures.