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12 15 (filledin).notebook September 22, 2016 12 Points, Lines, and Planes Objective: The students will understand basic terms and postulates of geometry and use them to form foundational knowledge of geometry. 1 12 15 (filledin).notebook September 22, 2016 Brief History of Geometry developed by Euclid, a Greek mathematician he developed axioms and theorems we still use today in his book, Elements Euclid's ideas were used until the 1800s geometry was then split into "Euclidean Geometry" and "NonEuclidean Geometry" Source: The History of Mathematics: An Introduction by David M. Burton 2 12 15 (filledin).notebook September 22, 2016 What is Geometry? geometry a mathematical system built on accepted facts, basic terms, and definitions undefined terms basic ideas that you can use to build the definitions of all the other figures in geometry 3 12 15 (filledin).notebook September 22, 2016 Undefined Terms point a location in space (It has NO size!) line an infinite number of points that extend in two opposite directions without end (It has NO thickness!) plane a flat surface that extends forever in all directions (It has NO thickness!) 4 12 15 (filledin).notebook September 22, 2016 Definitions collinear points that lie on the same line coplanar points or lines that lie in the same plane segment a part of a line that begins at one point and ends at another point ray a part of a line that begins at a point and extends infinitely in one direction 5 12 15 (filledin).notebook September 22, 2016 Definitions opposite rays two rays that share a common endpoint and form a line *intersect when geometric figures have one or more points in common *intersection the point(s) that geometric figures have in common 6 12 15 (filledin).notebook September 22, 2016 Postulates postulate (or axiom) a statement that is accepted as fact without proof Postulate 11 Through any two points there is exactly one line. Postulate 12 If two distinct lines intersect, then they intersect in exactly one point. 7 12 15 (filledin).notebook September 22, 2016 Postulates Postulate 13 If two distinct planes intersect, then they intersect in exactly one line. Postulate 14 Through any three noncollinear points there is exactly one plane. 8 12 15 (filledin).notebook September 22, 2016 13 Measuring Segments Objective: Today we will learn how to find and compare lengths of segments. 9 12 15 (filledin).notebook September 22, 2016 Ruler Postulate coordinate the real number that corresponds to a point on a number line distance between points A and B the absolute value of the difference of the coordinates a and b Formula: AB = |a – b| or |b – a| **Distance between the points is also the length of the segment AB 10 12 15 (filledin).notebook September 22, 2016 Segment Addition Postulate If A, B, and C are points on a number line and B is between A and C, then AB + BC = AC. 11 12 15 (filledin).notebook September 22, 2016 Congruent Segments congruent segments two or more segments that have the same length 12 12 15 (filledin).notebook September 22, 2016 Midpoint *midpoint a point that divides a segment into two congruent segments segment bisector a line, ray, or another segment that intersects a segment at its midpoint. **The line, ray, or segment is said to bisect the segment. 13 12 15 (filledin).notebook September 22, 2016 14 Measuring Angles Objective: Students are going to find and compare the measures of angles. 14 12 15 (filledin).notebook September 22, 2016 Angles angle a figure formed by two rays with a common endpoint sides of an angle the rays of an angle vertex of an angle the common endpoint of the two sides (rays) 15 12 15 (filledin).notebook September 22, 2016 Measuring an Angle measure of an angle COD the absolute value of the difference of the real numbers paired with OC and OD Formula: m COD = |c – d| or |d – c| 16 12 15 (filledin).notebook September 22, 2016 Types of Angles right angle an angle whose measure is 90o acute angle an angle whose measure is between 0o and 90o obtuse angle an angle whose measure is between 90o and 180o straight angle an angle whose measure is 180o reflex angle an angle whose measure is between 180o and 360o 17 12 15 (filledin).notebook September 22, 2016 Congruent Angles congruent angles two or more angles with the same measure **Written as OR 18 12 15 (filledin).notebook September 22, 2016 Angle Addition Postulate If B is a point in the interior of AOC, then the m AOB + m BOC = m AOC. 19 12 15 (filledin).notebook September 22, 2016 15 Exploring Angle Pairs Objective: We are going to learn how to identify special angle pairs and use their relationships to solve problems and justify results. 20 12 15 (filledin).notebook September 22, 2016 Types of Angle Pairs adjacent angles two coplanar angles with a common side, a common vertex, but NO common interior points *vertical angles two angles whose sides are opposite rays Note: Vertical angles are congruent. complementary angles two angles whose measures have a sum of 90o **Each angle is called the complement of the other. supplementary angles two angles whose measures have a sum of 180o **Each angle is called the supplement of the other. 21 12 15 (filledin).notebook September 22, 2016 Linear Pairs linear pair a pair of adjacent angles whose noncommon sides are opposite rays 22 12 15 (filledin).notebook September 22, 2016 Linear Pair Postulate If two angles form a linear pair, then the two angles are supplementary. 23 12 15 (filledin).notebook September 22, 2016 Concept Summary Page 35 24 12 15 (filledin).notebook September 22, 2016 Angle Bisector angle bisector a ray, segment, or line that divides an angle into two congruent angles 25