1.1 Points, Lines and Planes - PowerPoint PPT Presentation

1 / 46
About This Presentation
Title:

1.1 Points, Lines and Planes

Description:

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

Number of Views:443
Avg rating:3.0/5.0
Slides: 47
Provided by: JCHESTER
Category:

less

Transcript and Presenter's Notes

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
    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.

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
  • 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,
    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.

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.
Write a Comment
User Comments (0)
About PowerShow.com