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Transcript of Unit 1 Learning Outcomes 1: Describe and Identify the three undefined terms Learning Outcomes 2:...
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Unit 1
Learning Outcomes 1: Describe and Identify the three undefined terms
Learning Outcomes 2: Understand Angle Relationships
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Part 1
Definitions:
Points, Lines and Planes
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Undefined Terms
Points, Line and Plane are all considered to be undefined terms.– This is because they can only be explained
using examples and descriptions.– They can however be used to define other
geometric terms and properties
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Point– A location, has no shape or size– Label:
Line– A line is made up of infinite points and has no thickness or width, it will continue
infinitely.There is exactly one line through two points.– Label:
Line Segment– Part of a line– Label:
Ray– A one sided line that starts at a specific point and will continue on forever in one
direction.– Label:
< >A B
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Collinear – Points that lie on the same line are said to be
collinear – Example:
Non-collinear– Points that are not on the same line are said to be
non-collinear (must be three points … why?)– Example:
< >
F
A BE
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Plane– A flat surface made up of points, it has no depth
and extends infinitely in all directions. There is exactly one plane through any three non-collinear points
Coplanar– Points that lie on the same plane are said to be
coplanar
Non-Coplanar– Points that do not lie on the same plane are said to
be non-coplanar
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Intersect
The intersection of two things is the place they overlap when they cross. – When two lines intersect they create a
point.– When two planes intersect they create a
line.
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Space
Space is boundless, three-dimensional set of all points. Space can contain lines and planes.
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Practice Use the figure to give examples of the following:
Name two points.Name two lines.Name two segments.Name two rays.
Name a line that does not contain point T.Name a ray with point R as the endpoint.Name a segment with points T and Q as its endpoints.Name three collinear points.Name three non-collinear points.
QuickTime™ and a decompressor
are needed to see this picture.
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Part 2
Distance, Midpoint and Segments
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Distance Between Two Points
Distance on a number line • PQ = or
Distance on coordinate plane – The distance d between two points with
coordinates is given by
B−A A−B
x1, y1( )and x2 ,y2( )
d = x2 −x1( )2+ y2 −y1( )
2
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Examples
Example 1:– Find the distance between (1,5) and (-2,1)
Examples 2: – Find the distance between Point F and
Point B
-1-6< >
BE
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Congruent
When two segments have the same measure they are said to be congruent
Symbol:
Example:
≅
AB ≅ CD
< >
>< A B
C D
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Between
Point B is between point A and C if and only if A, B and C are collinear and
AB + BC =AC
< >A B C
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Midpoint
Midpoint– Halfway between the endpoints of the
segment. If X is the MP of then AB
AX =XB
< >XA B
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Finding The Midpoint
Number Line– The coordinates of the midpoint of a segment
whose endpoints have coordinates a and b is
Coordinate Plane– The coordinates of midpoint of a segment whose
endpoints have coordinates
are
a +b2
x1, y1( )and x2 ,y2( )x1 + x2
2,y1 + y2
2⎛⎝⎜
⎞⎠⎟
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Examples
The coordinates on a number line of J and K are -12 and 16, respectively. Find the coordinate of the midpoint of
Find the coordinate of the midpoint of
for G(8,-6) and H(-14,12).
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Segment Bisector
A segment bisector is a segment, line or plane that intersects a segment at its midpoint.
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Segment Addition Postulate
– if B is between A and C, then
AB + BC = AC
– If AB + BC = AC, then B is between
A and C
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Part 3
Angles
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Angle
An angle is formed by two non-collinear rays that have a common endpoint. The rays are called sides of the angle, the common endpoint is the vertex.
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Kinds of angles
Right Angle
Acute Angle
Obtuse Angle
Straight Angle / Opposite Rays
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Congruent Angles
Just like segments that have the same measure are congruent, so are angles that have the same measure.
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Angle Bisector
A ray that divides an angle into two congruent angles is called an angle bisector.
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Angle Addition Postulate
– If R is in the interior of <PQS, then m<PQR + m<RQS = m<PQS
– If m<PQR + m<RQS = m<PQS, then R is in the interior of <PQS
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Measuring Angles
How to use a protractor. – 1.) Line up the base line with one ray of
your angle. – 2.) Follow the base line out to zero, if you
are at 180 switch the protractor around.– 3.) Trace to protractor up until you reach
the second ray of your angle.– 4) The number your finger rests on is your
angle measure.
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Part 4
Angle Relationships
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Pairs of Angles
Adjacent Angles - are two angles that lie in the same plane, have a common vertex, and a common side, but no common interior points Vertical Angles-are two non-adjacent angles formed by two intersecting linesLinear Pair - is a pair of adjacent angles who are also supplementary
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Angle Relationships
Complementary Angles - Two angles whose measures have a sum of 90
Supplementary Angles - are two angles whose measures have a sum of 180
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Part 5
Angle Theorems
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Theorem 2.3
Supplement Theorem - – If two angles form a linear pair, then they
are supplementary angles
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Theorem 2.4
Complement Theorem– If the non-common sides of two adjacent
angles form a right angle, then the angles are complementary angles.
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Theorem 2.6
Angles supplementary to the same angle or to congruent angles are congruent
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Theorem 2.7
Angles complementary to the same angle or to congruent angles are congruent
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Theorem 2.8
Vertical Angles Theorem– If two angles are vertical, then they are
congruent
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Part 6
Perpendicular Lines and their theorems
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Perpendicular Lines
Lines that form right angles are perpendicular– Perpendicular lines intersect to form 4 right angles– Perpendicular lines form congruent adjacent
angles– Segments and rays can be perpendicular to lines
or to other line segments or rays– The right angle symbol in a figure indicates that
the lines are perpendicular.
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Theorems
Theorem 2.9 - Perpendicular lines intersect to form four right angles
Theorem 2.10 - All right angles are congruent
Theorem 2.11 - Perpendicular lines form congruent adjacent angles
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More Theorems
Theorem 2.12 - If two angles are congruent and supplementary, the each angle is a right angle
Theorem 2.13 - If two congruent angles form a linear pair, then they are right angles.
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Unit 1
The End!