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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.
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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
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1­2 ­ 1­5 (filled­in).notebook
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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
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1­2 ­ 1­5 (filled­in).notebook
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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!)
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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
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1­2 ­ 1­5 (filled­in).notebook
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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
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1­2 ­ 1­5 (filled­in).notebook
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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.
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1­2 ­ 1­5 (filled­in).notebook
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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.
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1­2 ­ 1­5 (filled­in).notebook
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1­3 Measuring Segments
Objective: Today we will learn how to find and compare lengths of segments.
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1­2 ­ 1­5 (filled­in).notebook
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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
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1­2 ­ 1­5 (filled­in).notebook
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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.
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1­2 ­ 1­5 (filled­in).notebook
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Congruent Segments
congruent segments ­ two or more segments that have the same length
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1­2 ­ 1­5 (filled­in).notebook
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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.
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1­2 ­ 1­5 (filled­in).notebook
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1­4 Measuring Angles
Objective: Students are going to find and compare the measures of angles.
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1­2 ­ 1­5 (filled­in).notebook
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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)
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1­2 ­ 1­5 (filled­in).notebook
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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|
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1­2 ­ 1­5 (filled­in).notebook
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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
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1­2 ­ 1­5 (filled­in).notebook
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Congruent Angles
congruent angles ­ two or more angles with the same measure
**Written as OR 18
1­2 ­ 1­5 (filled­in).notebook
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Angle Addition Postulate
If B is a point in the interior of AOC, then the m AOB + m BOC = m AOC.
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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.
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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.
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Linear Pairs
linear pair ­ a pair of adjacent angles whose non­common sides are opposite rays
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Linear Pair Postulate
If two angles form a linear pair, then the two angles are supplementary.
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Concept Summary ­ Page 35
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Angle Bisector
angle bisector ­ a ray, segment, or line that divides an angle into two congruent angles
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