Geometry - PowerPoint PPT Presentation

About This Presentation
Title:

Geometry

Description:

Identify and model points, lines, and planes. Identify collinear and coplanar points and intersecting lines and planes in space. ... Drawn with arrowhead at each end ... – PowerPoint PPT presentation

Number of Views:418
Avg rating:3.0/5.0
Slides: 112
Provided by: lnicol
Category:

less

Transcript and Presenter's Notes

Title: Geometry


1
Geometry
  • Ms. Toney

2
Orientation
  • Student Information Sheet
  • Classroom Rules
  • Classroom Policies
  • Calendar

3
What you will learn today
  • Identify and model points, lines, and planes.
  • Identify collinear and coplanar points and
    intersecting lines and planes in space.

4
Geometry in the Real World?
  • Where are there points, lines, and/or planes in
    this classroom?
  • What about outside the classroom, like in nature?

5
Some Important Definitions
  • Point
  • A location
  • Drawn as a dot
  • Named by a capital letter
  • Has no shape and no size
  • Example

6
Some Important Definitions
  • Line
  • Made up of points, has no thickness or width
  • Drawn with arrowhead at each end
  • Named by the letters representing two points on
    the line or a lower case script letter
  • There is exactly one line through any two points
  • Example
  • Collinear
  • Points on the same line

7
Some Important Definitions
  • Plane
  • A flat surface made up of points
  • Has no depth and extends infinitely in all
    directions
  • Drawn as a shaded figure
  • Named by a capital script letter or by the letter
    naming the three noncollinear points
  • There is exactly one plane through any three
    noncollinear points
  • Points are often used to name lines and planes
  • Example
  • Coplanar
  • Points that lie on the same plane

8
Example
  • Use the figure to name each of the following
  • A line containing point D
  • A plane containing point B

9
You Do It
  • Use the following figure to name each of the
    following
  • A line containing point K
  • A plane containing point L

10
Real World Examples
11
Undefined Terms
  • Point, line and plane are undefined terms
  • Have only been explained using examples and
    descriptions
  • We can still use these to define other geometric
    terms and properties
  • Two lines intersect at a point
  • Lines can intersect planes
  • Planes can also intersect each other

12
Example
  • Draw and label a figure
  • for each relationship
  • Lines GH and JK intersect
  • at L for G(-1, -3), H(2, 3),
  • J(-3, 2), and K(2, -3) on a
  • coordinate plane. Point M
  • is coplanar with these point
  • but not collinear with lines
  • GH or JK.
  • Line TU lies in a plane
  • Q and contains point R

13
Your Turn
  • Draw and label a figure
  • for each relationship
  • Line QR on a coordinate
  • plane contains Q(-2, 3)
  • and R(4, -4). Add point
  • T so that T is collinear
  • with these points
  • Plane R containing lines
  • AB and DE intersect at point
  • Add point C on plane R
  • so that it is not collinear
  • with lines AB or DE

14
Space
  • A boundless, three dimensional set of all
    points
  • Can contain lines and planes

15
Example
  • How many planes
  • appear in this figure?
  • Name three points
  • that are collinear.
  • Are points G, A, B,
  • and E coplanar? Explain.

16
Example
  • How many planes
  • appear in this figure?
  • Name three points
  • that are collinear.
  • Are points A, B, C,
  • and D coplanar? Explain.

17
Example
  • How many planes
  • appear in this figure?
  • Name three points
  • that are collinear.
  • Are points X, Y, Z,
  • and P coplanar? Explain.
  • At what point do lines
  • PR and TZ intersect?

18
Classwork
  • Worksheet

19
Homework
  • Workbook

20
(No Transcript)
21
What you will learn today
  • Measure segments.
  • Find the distance between two points.
  • Find the midpoint of a segment.

22
Units of Measure
  • When you see this sign, what unit of measure do
    you believe is being used?

23
Units of Measure
  • Actually in Australia, the unit of measure is
    kilometers.
  • Units of measure give us points of reference when
    evaluating the sizes of objects.

24
Measure Line Segments
  • Line Segment
  • Also called a segment
  • Can be measured because it has two endpoints
  • Named
  • The length or measure of is AB.
  • The length of a segment is only as precise as the
    smallest unit on the measuring device.

25
Example
  • Find AC.
  • Find DE.
  • Find y and PQ if P is between Q and R, PQ 2y,
    QR 3y 1, and PR 21.

26
Your Turn
  • Find LM.
  • Find XZ.
  • Find x and ST if T is between S and U, ST 7x,
    SU 45, and TU 5x 3.

27
More Terms
  • Congruent
  • When two segments have the same measure
  • Segments and angles are congruent
  • Distant and measures are equal

28
End of 1.2
29
Distance
  • Is always positive
  • Because you use whole numbers
  • Ways to find distance
  • Number line
  • Pythagorean Theorem
  • c2 a2 b2
  • Distance Formula

30
Example
  • Use the number lines to find the following

31
Example
32
Example
33
Your Turn
34
Midpoint
  • The point halfway between the endpoints of a
    segment
  • If B is the midpoint of the AB BC
  • Two formulas
  • Number Line
  • Coordinate Plane

35
Example
  • 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).
  • A is an endpoint and B is the midpoint located at
    A(3, 4) and B(-2, 1). Find the other endpoint C.

36
Your Turn
  • Find coordinates of D if E(-6, 4) is the midpoint
    of and F has coordinates (-5,
    -3).
  • What is the measure of if Q is the
    midpoint of ?

37
Segment Bisector
  • A segment, line or plane that intersects a
    segment at its midpoint

38
(No Transcript)
39
Quiz Time
  • Please clear off your desk
  • You will have 20 25 minutes to complete your
    quiz
  • When you are finished turn your quiz in and sit
    quietly in your sit until everyone has finished
  • We will begin todays lesson after the quiz

40
Angle Measure
  • Ray
  • Part of a line
  • Has one endpoint and extends indefinitely in one
    direction
  • Named stating the endpoint first and then any
    other point on the ray

41
More Terms
  • Angle
  • Formed by two noncollinear rays that have a
    common endpoint
  • Rays are called sides of the angle
  • Common endpoint is the vertex

42
  • An angle divides a plane into three distinct
    parts
  • Points A, D, and E lie on the angle
  • Points C and B lie in the interior of the angle
  • Points F and G lie in the exterior of the angle

43
Example
  • Name all the angles that have W as a vertex.
  • Name the sides of angle 1.
  • Write another name for angle WYZ.

44
(No Transcript)
45
Types of angles
  • Right Angle
  • Measurement of A 90

46
Types of Angles
  • Acute Angle
  • Measurement of B is less than 90

47
Types of angles
  • Obtuse angles
  • Measurement of C is less than 180 and greater
    than 90

48
Congruent Angles
  • Just like segments that have the same measure are
    congruent, angles that have the same measure are
    congruent.

49
Example
  • Wall stickers of standard shapes are often used
    to provide a stimulating environment for a young
    childs room. Find and
    ,
    , and

50
You Do It
  • A trellis is often used to provide a frame for
    vining plants. Some of the angles formed by the
    slats of the trellis are congruent angles. If

51
Bisectors
  • Segment Bisector
  • A segment, line or plane that intersects a
    segment at its midpoint
  • Angle Bisector
  • A ray that divides an angle into two congruent
    angles

52
(No Transcript)
53
Warm Up 3
  • Name the three different types of angles and
    describe them.
  • What is a bisector?

54
(No Transcript)
55
(No Transcript)
56
Today you have a choice!
  • Option 1
  • Section 1.5
  • Option 2
  • Review of 1.1 1.4 (lots of work)

57
Option 2 Review 1.1 1.4
  • Textbook
  • Pages 9 - 10
  • 13 -18, 21 26, 30 - 35
  • Page 17
  • 12 15, 22 26, 28 - 32
  • Pages25 - 26
  • 13 16, 23, 24, 31, 32, 37, 38, 43, 44
  • Pages 34 - 35
  • 12 -37, 50

58
(No Transcript)
59
Angle Relationships
  • Adjacent Angles
  • Two angles that lie in the same plane
  • Have a common vertex and a common side
  • Have no common interior angles

60
Angle Relationships
  • Vertical Angles
  • Two nonadjacent angles formed by two intersecting
    lines and they are congruent.

61
Angle Relationships
  • Linear Pair
  • A pair of adjacent angles whose noncommon sides
    are opposite rays

62
Example
  • Refer to the picture
  • Name an angle pair that satisfies each condition
  • Two obtuse vertical angles
  • Two acute vertical angles
  • Two angles that form a linear pair
  • Two acute adjacent angles

63
You Do It
  • Name an angle pair that satisfies each condition
  • Two obtuse vertical angles
  • Two acute adjacent angles
  • Name a segment that is perpendicular to segment
    FC.

64
Angle Relationships
  • Complementary Angles
  • Two angles whose have a sum of 90

65
Angle Relationships
  • Supplementary Angles
  • Two angles whose measures have a sum of 180

66
Example
  • Find the measures of two supplementary angles if
    the measure of one angle is 6 less than five time
    the measure of the other angle

67
You Do It
  • Find the measures of two complementary angles if
    the difference in the measures of the two angles
    is 12.

68
Angle Relationships
  • Perpendicular Lines
  • Lines that form right angles
  • Intersect to form congruent adjacent angles
  • Segments and rays can be perpendicular to lines
    or to other line segments and rays
  • - is read is perpendicular to and this symbol
    will indicate that two lines are perpendicular

69
Example
  • Find x so that

70
You Do It
  • Find x and y so that .

71
Things to Remember
  • While two lines may appear to be perpendicular in
    a figure, you cannot assume this is true unless
    other information is given.
  • The table on page 40 in your textbook has a list
    of things that may be assumed and things that may
    not be assumed.

72
(No Transcript)
73
Transparency 6
Click the mouse button or press the Space Bar to
display the answers.
74
Transparency 6a
75
(No Transcript)
76
(No Transcript)
77
Polygon
  • Is a closed figure formed by a finite number of
    coplanar segments such that
  • The sides that have a common endpoint are
    noncollinear
  • Each side intersects exactly two other sides, but
    only at their endpoints
  • Named by the letters of its vertices, written in
    consecutive order
  • Examples

78
Concave or Convex?
  • Polygons can be concave or convex.
  • How do we know which one?
  • Suppose the line containing each side is drawn.
  • If any of the lines contain any point in the
    interior of the polygon, then it is concave.
  • Otherwise it is convex.
  • Example

79
You are already familiar with many polygons such
as triangles, squares, and rectangles. A polygon
with n sides is an n gon. The table to the
right list some common names for various
categories of polygon. A convex polygon in which
all the sides are congruent and all the angles
congruent is called a regular polygon.
Number of Sides Polygon
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
12 Dodecagon
n n gon
80
Example
  • Name each polygon by its number of sides. Then
    classify it as convex or concave and regular or
    irregular.

81
Your Turn
  • Name each polygon by its number of sides. Then
    classify it as convex or concave and regular or
    irregular.

82
Perimeter
  • The sum of the lengths of its sides, which are
    segments.
  • Some shapes have special formulas, but they all
    come from the basic definition of perimeter.
  • Triangle P a b c
  • Square P s s s s 4s
  • Rectangle P l w l w 2l 2w

83
Example
  • A landscape designer is putting black plastic
    edging around a rectangular flower garden that
    has length 5.7 meters and width 3.8 meters. The
    edging is sold in 5-meter lengths.
  • Find the perimeter of the garden and determine
    how much edging the designer should buy.
  • Suppose the length and width of the garden are
    tripled. What is the effect on the perimeter and
    how much edging should the designer buy?

84
Your Turn
  • A masonry company is contracted to lay three
    layers of decorative brick along the foundation
    for a new house given the dimensions below.
  • Find the perimeter of the foundation and
    determine how many bricks the company will need
    to complete the job. Assume that one brick is 8
    inches long.
  • The builder realizes he accidentally halved the
    size of the foundation in part a, so he reworks
    the drawing with the correct dimensions. How
    will this affect the perimeter of the house and
    the number of bricks the masonry company needs?

85
Example using the Distance Formula
  • Find the perimeter of triangle PQR if P(-5, 1),
    Q(-1, 4), and R(-6, -8).
  • Find the perimeter of pentagon ABCDE with A(0,
    4), B(4, 0), C(3, -4), D(-3, -4), and E(-3, 1).

86
Your Turn
  • Find the perimeter of quadrilateral PQRS with
    P(-3, 4), Q(0, 8), R(3, 8), and S(0, 4).

87
Perimeter to Find Sides
  • The length of a rectangle is three times the
    width. The perimeter is 2 feet. Find the length
    of each side

88
Your Turn
  • The width of a rectangle is 5 less than twice its
    length. The perimeter is 80 centimeters. Find
    the length of each side.

89
Homework
  • Workbook
  • Lesson 1.6
  • 1 - 13

90
(No Transcript)
91
(No Transcript)
92
(No Transcript)
93
(No Transcript)
94
(No Transcript)
95
(No Transcript)
96
(No Transcript)
97
(No Transcript)
98
(No Transcript)
99
(No Transcript)
100
(No Transcript)
101
(No Transcript)
102
(No Transcript)
103
(No Transcript)
104
(No Transcript)
105
(No Transcript)
106
(No Transcript)
107
(No Transcript)
108
(No Transcript)
109
(No Transcript)
110
(No Transcript)
111
(No Transcript)
Write a Comment
User Comments (0)
About PowerShow.com