Download Section 2.1 – Undefined terms, postulates, segments and angles

Mth 97
Winter 2013
Section 2.1
Problem Solving (Chapter 1)
Polya’s 4 Steps: 1) Understand the problem; 2) Devise a plan; 3) Carry out the plan; 4) Look Back
Problem Solving Strategies
1. Draw a Picture (See pages 5 and 6 of your text)
– Work problem 2 on page 10.
2. Guess and Test (pages 6 – 7)
– Work problem 17 on page 12 using toothpicks.
3. Use a Variable (pages 8 – 9) – Work the following problem.
The perimeter of a picture is 26 inches. Its length is 3 inches greater than its width. What are the
dimensions of the picture? (Hint: Define a variable, then write and solve an equation.)
Section 2.1 – Undefined terms, postulates, segments and angles
Axiomatic System: Undefined terms → Definitions → Postulates → Theorems
Undefined terms: Can be described but cannot be given precise definitions
Point
Line
Space: The set of ________ points
Plane
Geometric Figure: Any collection of _____________
Collinear Points: Points that line on the same ____________.
Non-collinear Points: Points that are ________ all on the same line.
Coplanar Points: Points that lie in the same ________________.
Non-coplanar Points: Points that are_______ contained in the same plane.
Postulates: Statements that we assume to be ___________which state relationships between defined
and undefined terms.
Theorems: Results deduced from undefined terms, definitions or postulates and can be _____________
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Mth 97
Winter 2013
Section 2.1
Relationships Among Points, Lines, and Planes
Postulate 2.1 – Every line contains at least two distinct points.
Postulate 2.2 – Two points are contained in one and only one line.
Postulate 2.3 – If two points are in a plane the line containing these points is also in the plane.
Postulate 2.4 – Three non-collinear points are contained in one and only one plane and every plane
contains at least three non-collinear points.
Postulate 2.5 – In space, there exist at least four points that are not all coplanar.
Line Segment: determined by its endpoints & includes all the points __________ these points on a line
Postulate 2.6 (The Ruler Postulate)
– Every line can be made into an exact copy of the real number line using 1–1 correspondence.
A
B
C
D
E
F
G
H
I
J
-4
-3
-2
-1
0
1
2
3
4
5
The distance between two points is the ___________________ difference between their coordinates.
AE =
BF =
DH =
Congruent segments are segments that have the same______________________.
AB  DE because AB = DE = 1.
CF 
CF 
CF 
because CF = ______ = ______ = _______ = ______
If the coordinates of A, B, and C are respectively 3 5 , 2 5 , and  5 , find each segment length.
AB =
BC =
AC =
A ray is a portion of a line that has one endpoint and extends indefinitely in _________ direction and is
named by the endpoint followed by a point on the ray written below a ray symbol,
A
B
C
D
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Mth 97
Winter 2013
Section 2.1
Angles and Their Measure
Angle: Formed by two line segments or rays meeting at a common endpoint.
Vertex ______
Sides
_________ and _________
Named with an angle symbol and its vertex _____________
or an angle symbol and three points (side-vertex-other side)
_____________ or ____________
Postulate 2.7 (The Protractor Postulate)
If we place one ray of an angle at 0° on a protractor and we place the vertex at the midpoint of the
bottom edge (or center crosshairs), then there is a 1-1 correspondence between all other rays that can
serve as the second side of the angle and the real numbers between 0° and 180°. This number is called
the “measure” of the angle.
Measure the angle above and then do number 1 on today’s ICA (In-class Assignment).
Types of Angles
Acute– has a measure
Right – has a measure of 90°
between 0° and 90°
Straight – has a measure of 180°
Obtuse – has a measure
between 90° and 180°
Reflex – has a measure between 180° and 360°
Congruent angles have the same ________________________
If the measure of DEF is 30° and DEF  GHI , then the measure of GHI is ____________
Complementary – two angles whose sum is 90°
Supplementary – two angles whose sum is 180°
Do number 2 on today’s ICA.
Adjacent angles – have a common vertex and side but no common interior points.
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Mth 97
Winter 2013
Section 2.1
mAGB  75
Given
mCGE  90
C
B
mCGD  50
mFGC  175
mDGE 
Find
A
mFGE 
D
G
mAGF 
mBGC 
In the drawing above name pairs of:
Complementary angles
E
F
Supplementary angles
Adjacent angles
More precise measures of angles include fractions of degrees or minutes and seconds where
60 minutes = 1 degree and 60 seconds = I minute.
Use dimensional analysis to express 32°15’40” in degrees as a decimal rounded to thousandths.
Use dimensional analysis to express 15.12°
in degrees and minutes.
Use dimensional analysis to express 75.3°
in degrees minutes and seconds.
Complete the rest of ICA 1.
One radian is the measure of the central angle of a circle that cuts off a portion of the distance around
the circle that is the same length as the radius of the circle. The radian measure of a full circle is 2π, so
2π = 360° or for half a rotation, π = 180°. Use dimensional analysis to find the radian measure for the
following angles.
15°
120°
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