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1.1 Points, Lines and Planes

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1.1 Points, Lines and Planes Undefined Terms There are three undefined terms in Geometry. They are Points, Lines and Planes. They are considered undefined because ... – PowerPoint PPT presentation

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Title: 1.1 Points, Lines and Planes

1
1.1 Points, Lines and Planes
2
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.

3
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

5
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
6
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
7
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)

8
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)

9
Describing What you see!
• There are key terms such as
• Lies in,
• Contains,
• Passes through,
• Intersection,
• See Pg 12.

10
1.2 Linear Measure and Precision
11
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.

12
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.
13
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.
14
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!
15
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
16
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.

17
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.

18
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.
19
1.3 Distance and Midpoints
20
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.

21
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.

22
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
23
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.

24
Distance (Shortcut)
(8, 10)
5
Find the distance between these two points.
6
(2, 5)
Or use the Pythagorean theorem.
Create a right triangle.
d2 62 52 36 25 61 so d v61
25
1.4 Angle Measure
26
Another Portion of a Line
• We already talked about segments, now let us talk
• 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.

27
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
28
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.

29
Interior vs. Exterior
Exterior
Interior
Exterior
Exterior
30
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

31
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.

32
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.

33
1.5 Angle Relationships
• Angle Pairs

34
• 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?
35
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.
36
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!
37
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.
38
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.
39
Perpendicular Lines
• Perpendicular Lines intersect to form four right
angles.
• Perpendicular Lines intersect to form congruent,
• 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.

40
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.

41
1.6 Polygons
42
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.

43
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
44
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.

45
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

46
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.