1.1 Points, Lines and Planes

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1.1 Points, Lines and Planes
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Undefined Terms

• There are three undefined terms in Geometry.
• They are Points, Lines and Planes.
• They are considered undefined because they have
  only been explained using examples and
  descriptions.

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Points

• Points are simply locations.
• Drawn as a dot.
• Named by using a Capital Letter
• No size or shape.
• Verbally you say Point P

P
4
Line
l

• A line is a collection of an infinite number of
  points (named or un-named).
• Points that lie on the line are called Collinear.
• Collinear Points are points that are on the same
  line.
• Draw a line with arrows on each end to signify
  that it is infinite in both directions.
• Name by either two points on the line or lower
  case script letter

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Line (Continued)

• A line has only one dimension (length).
• It has no width or depth.
• Postulate There exists exactly one line through
  two points.
• To plot a point on a number line, youll need
  only one number.

l
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Plane

• A plane is a flat surface made up of an infinite
  number of points.
• Points that lie on the same plane are said to be
  Coplanar.
• Planes are named by using a capital, script
  letter or three non-collinear points.

Plane RFK
Plane P
P
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Plane (Continued)

• Although a plane looks like it is a
  quadrilateral, it is in fact infinitely long and
  wide.
• Planes (Coordinate Plane) have two dimensions
  so you need two numbers to plot a point. P(x,y)

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Space

• Space is a boundless, three dimensional set of
  all points. Space can contain, points, lines and
  planes.
• In chapter 13 you will see that youll need three
  numbers to plot a point in space. P(x,y,z)

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Describing What you see!

• There are key terms such as
• Lies in,
• Contains,
• Passes through,
• Intersection,
• See Pg 12.

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1.2 Linear Measure and Precision
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Introduction

• Lines are infinitely long.
• There are portions of lines that are finite. In
  other words, they have a length.
• The portion of a line that is finite is called a
  Line Segment.
• A line segment or segment has two distinct end
  points.

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Betweenness

• Betweenness of points is the relationships among
  three collinear points.
• We can say B is between A and C and you should
  think of this picture.

C
A
B
Notice that B is between but not in exact middle.
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Example
Find the length of LN or LN?
From this picture we can always write this
equation LM MN LN.
So, if LM 3 and MN 5, we can say that LN 8.
What if LM 2y, MN 21 and LN 3y1?
Then we can write.. 2y 21 3y 1
From this equation we can solve for y and
substitute that value to find LN.
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Congruence of Segments

• Segments can be Congruent if they have the same
  measurement.
• We have a special symbol for congruent. It is an
  equal sign with a squiggly line above it.

Hint Shapes can be congruent, measurements can
only be equal. So if youre talking about a
shape, you say congruent or not congruent!
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Congruence

• Congruence can not be assumed!
• Dont think, that just because it looks like the
  same length, it is.
• Short cut we can use congruent marks to show
  that segments are congruent.

C
Q
A
P
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Precision (H)

• The precision of a measurement depends on the
  smallest unit of measure available on the
  measuring tool.
• The precision will always be ½ the smallest unit
  of measure of the measuring device.

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Precision (H)

• Here to find the length we would have to say it
  is four units long b/c it is closer to 4 than 5.

• The precision of this measuring device is ½ the
  smallest unit of measure, 1, or the precision is
  1/2.

• We can say the measurement is 4 1/2

• So the segment could be as small as 3 ½ or as
  big as 4 ½ and still be called 4.

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Precision (Cont)

• Here we have the same segment but a different,
  more accurate measuring device.

• The units are broken down into ¼s. The segment
  is closer to 4 ¼ than 4 ½.

• The precision is ½ of ¼, or 1/8th.

• So the length is 4 ¼ 1/8th.

Smallest 4 1/8th Largest 4 3/8th.
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1.3 Distance and Midpoints
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Distance

• The coordinates of the two endpoints of a line
  segment can be used to find the length of the
  segment.
• The length from A to B is the same as it is from
  B to A.
• Thus AB BA (This stands for the measurement of
  the segment)
• Distance (length) can never be negative.

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Midpoint

• Definition - The midpoint of a segment is the
  point ½ way between the endpoints of the segment.
• If B is the Midpoint (MP) of
• then, AB BC.
• The midpoint is a location, so it can be positive
  or negative depending on where it is.

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One Dimensional
B
C
D
A
-3 -2 -1 0 1 2 3 4

• If point A was at -3 and point B was at 2, then
  AB5 b/c the formula for AB A B

• The MP formula is (AB)/2

(-32)/2 -1/2

• What if point C was at -2 and D was at 4,
• what is CD?

CD 4 (-2) or -2 4 6
MP is (4 (-2))/2 1
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Two Dimensional

• We designate points on a plane using ordered
  pair P(x,y).
• We plot them on the Cartesian Coordinate plane
  just as you did in Alg I.
• Again, distances can not be negative because
  lengths are not negative.
• Midpoints can be either positive or negative b/c
  it is simply a location.

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Distance (Shortcut)
(8, 10)
5
Find the distance between these two points.
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(2, 5)
Or use the Pythagorean theorem.
Create a right triangle.
d2 62 52 36 25 61 so d v61
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1.4 Angle Measure
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Another Portion of a Line

• We already talked about segments, now let us talk
  about Rays.
• A ray is a portion of a line that has only one
  end point. It is infinite in the other
  direction.
• A ray is named by using the end point and any
  other point on the ray.

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Opposite Rays

• If you chose a point on a line, that point
  determines exactly two rays called Opposite Rays.
• These two opposite rays form a line and are said
  to be collinear rays.

C
A
B
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Angles

• Angles are created by two non-collinear rays
  that share a common end point.

ltCED or ltDEC

• Angles are named by using one letter from one
  side, the vertex angle, and one letter from the
  other side.

• An angle consists of two sides which are rays
  and a vertex which is a point.

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Interior vs. Exterior
Exterior
Interior
Exterior
Exterior
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Classifications of Angles

• Right Angle An angle with a measurement of
  exactly 90 mltABC90
• Acute Angle An angle with a measurement more
  than 0 but less than 90 0 lt mltABC lt 90
• Obtuse Angle An angle with a measurement more
  than 90 but less than 180 90 lt mltABC lt 180

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Congruence of Angles

• Angles with the same measurement are said to be
  congruent.

• mltACE 25 and mltDCG 25 since the two
  angles have the same measurement we can say that
  theyre congruent.

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Angle Bisector

• An angle bisector is a Ray that divides an angle
  into two congruent angles.

P

• If is an angle bisector.
• Then ltADP is congruent to ltPDH.

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1.5 Angle Relationships

• Angle Pairs

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Adjacent 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.

ltABC and ltCBD are Adjacent Angles. They dont
have to be equal.
Common Side?
Common Vertex?
B
No Common Interior Point?
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Vertical Angles

• Vertical Angles Are two non-adjacent angles
  formed by intersecting lines.

• Two Intersecting Lines?

ltABD and ltCBE are non-adjacent angles formed by
intersecting lines. They are Vertical Pair.
What else?
ltABC and ltDBE are also Vertical Pair.
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Linear Pair

• Linear Pair Is a pair of adjacent angles whose
  non-common sides are opposite rays.

• Are ltLMP and PMN are Adjacent?

Yes!

• Are Rays ML and MN the Non-Common Sides?

Yes!

• Are Rays ML and MN Opposite Rays?

Yes!
ltLMP and ltPMN are Linear Pair!
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Complementary Angles

• Complementary Angles Are two angles whose
  measures have a sum of 90
• Do you see the word Adjacent in the definition?

No!
1
2
lt1 and lt2 are Comp.
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Supplementary Angles

• Supplementary Angles Are two angles whose
  measures have a sum of 180
• Do you see the word Adjacent in the definition?

No!
1
2
lt1 and lt2 are Supp.
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Perpendicular Lines

• Perpendicular Lines intersect to form four right
  angles.
• Perpendicular Lines intersect to form congruent,
  adjacent angles.
• Segments and rays can be perpendicular to lines
  or to other line segments or rays.
• The right angle symbol indicates that the lines
  are perpendicular.

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Assumptions

• Things that can be assumed.
• Coplanar, Intersections, Collinear, Adjacent,
  Linear Pair and Supplementary
• Things that can not be assumed.
• Congruence, Parallel, Perpendicular, Equal, Not
  Equal, Comparison.

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1.6 Polygons
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Polygon

• Polygon A closed figure whose sides are all
  segments and they only intersect at the end
  points of the segments.
• Polygons are named by using consecutive points at
  the vertices.
• Example A triangle with points of A, B and C is
  named ?ABC.

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Concave vs. Convex

• Concave A polygon is concave when at least one
  line that contains one of the sides passes
  through the interior.
• Convex A polygon is convex when none of the
  lines that contains sides passes through the
  interior.

Concave
Convex
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Classification by Sides

• Polygons are classified by the number of sides it
  has.
• 3 Triangle 4 Quadrilateral
• 5 Pentagon 6 Hexagon
• 7 Heptagon 8 Octagon
• 9 Nonagon 10 Decagon
• 11 Undecagon 12 Dodecagon
• Any polygon more than 12 then N-Gon. Example
  24 sides is a 24-gon.

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Regular Polygon

• Regular Polygon Is a polygon that is
  equilateral (all sides the same length),
  equiangular (all angles the same measurement) and
  convex.
• Examples
• Triangles Equilateral Triangle
• Quadrilateral - Square

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Perimeter

• Perimeter The sum of the lengths of all the
  sides of the polygon.
• May have to do distance formula for coordinate
  geometry problem.
• See example 3.