Geometry

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Geometry

• Ms. Toney

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Orientation

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

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What you will learn today

• Identify and model points, lines, and planes.
• Identify collinear and coplanar points and
  intersecting lines and planes in space.

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Geometry in the Real World?

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

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Some Important Definitions

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

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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

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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

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Example

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

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You Do It

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

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Real World Examples
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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

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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

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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

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Space

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

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Example

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

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Example

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

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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?

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Classwork

• Worksheet

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Homework

• Workbook

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What you will learn today

• Measure segments.
• Find the distance between two points.
• Find the midpoint of a segment.

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Units of Measure

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

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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.

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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.

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Example

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

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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.

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More Terms

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

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End of 1.2
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Distance

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

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Example

• Use the number lines to find the following

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Example
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Example
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Your Turn
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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

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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.

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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 ?

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Segment Bisector

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

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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

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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

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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

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• 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

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Example

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

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Types of angles

• Right Angle
• Measurement of A 90

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Types of Angles

• Acute Angle
• Measurement of B is less than 90

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Types of angles

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

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Congruent Angles

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

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Example

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

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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

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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

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Warm Up 3

• Name the three different types of angles and
  describe them.
• What is a bisector?

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Today you have a choice!

• Option 1
• Section 1.5
• Option 2
• Review of 1.1 1.4 (lots of work)

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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

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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

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Angle Relationships

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

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Angle Relationships

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

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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

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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.

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Angle Relationships

• Complementary Angles
• Two angles whose have a sum of 90

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Angle Relationships

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

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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

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You Do It

• Find the measures of two complementary angles if
  the difference in the measures of the two angles
  is 12.

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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

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Example

• Find x so that

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You Do It

• Find x and y so that .

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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.

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Transparency 6
Click the mouse button or press the Space Bar to
display the answers.
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Transparency 6a
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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

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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

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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
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Example

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

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Your Turn

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

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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

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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?

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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?

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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).

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Your Turn

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

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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

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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.

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Homework

• Workbook
• Lesson 1.6
• 1 - 13

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