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Unit 2. Basic Geometric Elements. Lesson 2.1. Points, Lines, and Planes. Lesson 2.1 Objectives. Define and write notation of the following: (G1.1.6) Point Line Plane Ray Line segment Collinear Coplanar End point Initial point Opposite rays Intersection.

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unit 2

Unit 2

Basic Geometric Elements

lesson 2 1

Lesson 2.1

Points, Lines, and Planes

lesson 2 1 objectives
Lesson 2.1 Objectives
  • Define and write notation of the following: (G1.1.6)
    • Point
    • Line
    • Plane
    • Ray
    • Line segment
    • Collinear
    • Coplanar
    • End point
    • Initial point
    • Opposite rays
    • Intersection
start up
Give your definition of the following:

Point

Line

These terms are actually said to be undefined, or have no formal definition.

However, it is important to have a general agreement on what each word means.

Start-Up
point
Point
  • A point has no dimension, it is merely a location.
    • Meaning it takes up no space.
  • It is usually represented as a dot.
  • When labeling we designate a capital letter as a name for that point.
    • We may call it Point A.

A

slide6

B

A

Line
  • A line extends in one dimension.
    • Meaning it goes straight in either a vertical, horizontal, or slanted fashion.
  • It extends forever in two directions.
  • It is represented by a line with an arrow on each end.
  • When labeling, we use lower-case letters to name the line.
    • Or the line can be named using two points that are on the line.
    • So we say Line n, or

AB

n

plane

A

C

B

Plane

M

  • A plane extends in two dimensions.
    • Meaning it stretches in a vertical direction as well as a horizontal direction at the same time.
  • It also extends forever.
  • It is usually represented by a shape like a tabletop or a wall.
  • When labeling we use a bold face capital letter to name the plane.
    • Plane M
    • Or the plane can be named by picking three points in the plane and saying Plane ABC.
collinear

A

B

C

Collinear
  • The prefix co- means the same, or to share.
  • Linearmeans

line.

  • So collinear means that points lie on the sameline.

We say that points A, B, and C are collinear.

coplanar

A

C

B

Coplanar
  • Coplanar points are points that lie on the same plane.

M

So points A, B, and C are said to be coplanar.

line segment

B

A

Line Segment
  • Consider the line AB.
    • It can be broken into smaller pieces by merely chopping the arrows off.
  • This creates a line segment or segment that consists of endpointsA and B.
    • This is symbolized as

AB

slide11

B

A

Ray
  • A ray consists of an initial point where the figure begins and then continues in one direction forever.
    • It looks like an arrow.
  • This is symbolized by writing its initial point first and then naming any other point on the ray, .
    • Or we can say rayAB.

AB

betweenness

A

B

C

Betweenness
  • When three points lie on a line, we can say that one of them is between the other two.
    • This is only true if all three points are collinear.
    • We would say that B is between A and C.
opposite rays
Opposite Rays
  • If C is between A and B on a line, then ray CA and ray CB are oppositerays.
    • Oppositerays are only opposite if they are collinear.

C

B

A

intersections of lines and planes

A

m

n

Intersections ofLines and Planes
  • Two or more geometric figures intersect if they have one or more points in common.
    • If there is no point or points shown, they the figures do not intersect.
    • The intersection of the figures is the set of points the figures have in common.
  • Two lines intersect at onepoint.
  • Two planes intersect at oneline.
example 2 2
Example 2.2

Answer the following

  • Name 3 points that are collinear.
    • C, B, D
  • Name 3 points that are not collinear.
    • ex: A, B, E or A, B, C
  • Name 3 points that are coplanar.
    • ex: A, B, E or B, C, D or B, C, E
  • Name 4 points that are not coplanar.
    • ex: A, B, E, C
  • What are two ways to name the plane?
    • Plane ABE or Plane F
  • What are two names for the line that passes through points C and B.
    • line g or
homework 2 1
Homework 2.1
  • Lesson 2.1 – Point, Line, Plane
    • p1-2
  • Due Tomorrow
lesson 2 2

Lesson 2.2

Distance,

Midpoint, and

Segment Addition

lesson 2 2 objectives
Lesson 2.2 Objectives
  • Utilize the distance formula. (G1.1.3)
  • Apply the midpoint formula. (G1.1.5)
  • Justify the construction of a midpoint. (G1.1.5)
  • Utilize the segment addition postulate. (G1.1.3)
  • Identify the symbol and definition of congruent. (G1.1.3)
  • Define segment bisector. (G1.1.3)
postulate 1 ruler postulate

E

C

D

Postulate 1: Ruler Postulate
  • The points on a line can be matched to real numbers called coordinates.
  • The distance between the points, say A and B, is the absolute value of the difference of the coordinates.
    • Distanceis always positive.

A

B

length
Length
  • Finding the distance between points A and B is written as
    • AB
  • Writing AB is also called the length of line segment AB.
postulate 2 segment addition postulate

A

B

C

Postulate 2: Segment Addition Postulate
  • If B is between A and C, then
    • AB + BC = AC.
  • Also, the opposite is true.
    • If AB + BC = AC, then B is between A and C.

BC

AB

AC

example 2 3

D

F

E

M

P

N

Example 2.3
  • Sketch and write the segment addition postulate if point E is between points D and F.

DE

+

EF

=

DF

Sketch and write the segment addition postulate if point M is between points N and P.

NM

+

MP

=

NP

example 2 4
Example 2.4

Find

  • GJ
    • GJ = 16
  • KM
    • KM = 36
  • XY
    • 71-29

XY = 42

  • LM
    • x + 2x = 18

3x = 18

x = 6

LM = 6

distance formula
Distance Formula

To find the distance on a graph between two points

A(1,2)

B(7,10)

We use the Distance Formula

AB =

(x2 – x1)2 + (y2 – y1)2

Distance can also be found using the Segment Addition Postulate, which simply adds up each segment of a line to find the total length of the line.

congruent segments
Congruent Segments
  • Segments that have the same length are called congruent segments.
    • This is symbolized by .

Hint: If the symbols are there, the congruent sign should be there.

If you want to state two segments are congruent, then you write

If you want to state two lengths are equal, then you write

example 2 5
Example 2.5

(x2 – x1)2 + (y2 – y1)2

Find the distance of each segment and identify if any of the segments are congruent.

  • J(1,1)K(0,5)

A(4,3)B(-1,6)

D(2,-3)E(-2,0)

L(2,1)M(-2,0)

midpoint

O

J

Y

Midpoint
  • The midpoint of a segment is the point that divides the segment into two congruent segments.
    • The midpointbisects the segment, because bisect means to divide into two equal parts.

We say that O is the midpoint

of line segment JY.

midpoint formula

A(1,2)

B(7,10)

(

)

,

Midpoint Formula

We can also find the midpoint of segment AB by

using its endpoints in…

The Midpoint Formula

Midpoint of AB =

(y1 + y2)

(x1 + x2)

2

2

This gives the coordinates of the midpoint, or point that is halfway between A and B.

example 2 6

(y1 + y2)

(x1 + x2)

2

2

(

)

,

Example 2.6

Find the midpoint

  • R(3,1)S(3,7)

T(2,4)S(6,6)

finding the other end

E

M

?

Finding the Other End
  • Many may say finding the midpoint is easy!
    • It is simply the average of the two endpoints.
  • Now imagine knowing the midpoint, one endpoint, and trying to find the coordinates of the other endpoint.
      • Try to remember what the midpoint formula does and work it backwards.
    • So here is what we are going to do:
      • Double the coordinates of the midpoint.
      • Subtract the coordinates of the known endpoint.
example 2 7
Example 2.7

Find the other endpoint given one endpoint, E, and the midpoint, M.

  • E(0,5)M(3,3)

E(-1,-3)M(5,9)

segment bisector

T

H

O

J

Y

Segment Bisector
  • A segment bisector is a segment, ray, line, or plane that intersects the original segment at its midpoint.
example 2 8
Example 2.8

Use the diagram to find the given measure if line l is a segment bisector.

109 in

ST = ½(109 in)

ST = 54.5 in

homework 2 2
Homework 2.2
  • Lesson 2.2 – Line Segments
    • p3-4
  • Due Tomorrow
lesson 2 3

Lesson 2.3

Angles and Their Measures

lesson 2 3 objectives
Lesson 2.3 Objectives
  • Identify more than one name for an angle. (G1.1.6)
  • Identify angle measures. (G1.1.6)
  • Classify angles as right, obtuse, acute, or straight. (G1.1.6)
  • Apply the angle addition postulate. (G1.1.3)
  • Utilize angle vocabulary to solve problems. (G1.1.6)
  • Define angle bisector and its uses. (G1.1.3)
what is an angle
What is an Angle?
  • An angle consists of two different rays that have the same initial point.
  • The rays form the sides of the angle.
  • The initial point is called the vertex of the angle.
    • Vertex can often be thought of as a corner.
naming an angle
Naming an Angle
  • All angles are named by using three points
    • First, name a point that lies on one side of the angle.
    • Second, name the vertex next.
      • The vertex is always named in the middle.
    • Finally, name a point that lies on the oppositeside of the angle.

So we can call It WON

Or NOW

W

N

O

using a protractor
Using a Protractor
  • To measure an angle with a protractor, do the following:
  • Place the cross-hairs of the protractor on the vertex of the angle.
  • Line up one side of the angle with the 0o line near the bottom of the protractor.
  • Read the protractor for the where the other side of the angle points.
example 2 9
Example 2.9

Protractor Stations

congruent angles
Congruent Angles
  • Congruent angles are angles that have the same measure.
    • To show that we are finding the measure of an angle…
      • Place a “m” before the name of the angle.

Congruent Angles

Equal Measures

example 2 10
Example 2.10

Give another name for the angle in the diagram above. Then, tell whether the angle appears to be acute, obtuse, right, or straight.

  • JKN
    •  NKJ,  K
      • right
  • KMN
    •  NMK
      • straight
  • PQM
    •  MQP
      • acute
  • JML
    •  LMJ
      • acute
  • PLK
    •  KLP
      • obtuse
other parts of an angle
Other Parts of an Angle
  • The interior of an angle is defined as the set of points that lie between the sides of the angle.
  • The exteriorof an angle is the set of points that lie outside of the sides of the angle.

Exterior

Interior

postulate 4 angle addition postulate
Postulate 4: Angle Addition Postulate
  • The Angle Addition Postulate allows us to add each smaller angle together to find the measure of a larger angle.

What is the total?

49o

32o

17o

example 2 11
Example 2.11

Use the given information to find the indicated measure.

3x + 15 + x + 7 = 94

4x + 22 = 94

4x = 72

25o

x = 18

3x + 1 + 2x – 6 = 135

5x – 5 = 135

85o

3(28) + 1

5x = 140

84 + 1

x = 28

85

adjacent angles

C

T

R

A

Adjacent Angles
  • Two angles are adjacent angles if they share a common vertex and side, but have no common interior points.
    • Basically they should be touching, but not overlapping.

CAT and TAR areadjacent.

CAR and TAR

are notadjacent.

angle bisector
Angle Bisector
  • An angle bisector is a ray that divides an angle into two adjacent angles that are congruent.
    • To show that angles are congruent, we use congruence arcs.
example 2 12
Example 2.12

8x – 16 = 4x + 20

4x = 3x + 6

5x – 11 = 4x + 1

4x – 16 = 20

x = 6

x – 11 = 1

4x = 36

x = 12

x = 9

2(4(9) + 20)

4(6) + 3(6) + 6

5(12) – 11 + 4(12) + 1

2(56)

60 – 11 + 48 + 1

24 + 18 + 6

mABC = 48o

mABC = 98o

mABC = 112o

homework 2 3
Homework 2.3
  • Lesson 2.3 - Angles and Their Measures
    • p5-7
  • Due Tomorrow
lesson 2 4

Lesson 2.4

Angle Pair Relationships

lesson 2 4 objectives
Lesson 2.4 Objectives
  • Identify vertical angle pairs. (G1.1.1)
  • Identify linear pairs. (G1.1.1)
  • Differentiate between complementary and supplementary angles. (G1.1.1)
vertical angles

1

2

4

3

Vertical Angles
  • Two angles are vertical angles if their sides form two pairs of opposite rays.
    • Basically the two lines that form the angles are straight.
  • To identify the vertical angles, simply look straight across the intersection to find the angle pair.
    • Hint: The angle pairs do not have to be vertical in position.
  • Vertical Angle pairs are always congruent!

1

2

4

3

linear pair
Linear Pair
  • Two adjacent angles form a linear pair if their non-common sides are opposite rays.
    • Simply put, these are two angles that share a straight line.
      • Just like neighbors share a fence line, but they must live on the same side of the road.
  • Since they share a straight line, their sum is…
    • 180o

1

2

example 2 13
Example 2.13

Find the measure of all unknown angles, when m1 = 57o.

  • m2
    • 123o
  • m3
    • 57o
  • m4
    • 123o
example 2 14

16y - 27 = 13y

Example 2.14

-27 = -3y

y = 9

Solve for x and y.

7x = 5x + 18

3x + 48 = 180

9x + 7 = 5x + 67

2x = 18

3x = 132

4x + 7 = 67

x = 9

x = 44

4x = 60

x = 15

complementary v supplementary
Complementary anglesare two angles whose sum is 90o.

Complementary angles can be adjacent or non-adjacent.

Supplementary anglesare two angles whose sum is 180o.

Supplementary angles can be adjacent or non-adjacent.

Complementary v Supplementary
example 2 15

m

n

Example 2.15

Find the measure of all unknown angles, given that m and n form a right angle and the m1 = 22o and 1  4.

  • m2
    • 68o
  • m5
    • 22o
  • m6
    • 68o
  • m4
    • 22o
  • m3
    • 68o
  • m7
    • 68o
  • m8
    • 22o
example 2 16
Example 2.16

A and B are complementary. Find mA and mB.

  • mA = 2x + 12mB = 9x – 10
    • 2x + 12 + 9x – 10 = 90

11x + 2 = 90

11x = 88

x = 8

A and B are supplementary. Find mA and mB.

  • mA = 12x + 32mB = 4x – 12
    • 12x + 32 + 4x – 12 = 180

16x + 20 = 180

16x = 160

x = 10

mA = 2(8) + 12

mB = 9(8) - 10

mA = 12(10) + 32

mB = 4(10) - 12

mA = 16 + 12

mB = 72 - 10

mA = 120 + 32

mB = 40 - 12

mA = 28o

mB = 62o

mA = 152o

mB = 28o

perpendicular lines

b

a

Perpendicular Lines
  • When two lines intersect to form a right angle, they are said to be perpendicular lines.
homework 2 4
Homework 2.4
  • Lesson 2.4 - Angle Pair Relationships
    • p7-8
  • Due Tomorrow
lesson 2 5

Lesson 2.5

Introduction to Parallel Lines

and

Transversals

lesson 2 5 objectives
Lesson 2.5 Objectives
  • Identify angle pairs formed by a transversal. (G1.1.2)
  • Compare parallel and skew lines. (G1.1.2)
lines and angle pairs
Lines and Angle Pairs

Alternate Exterior Angles – because they lie outside the two lines and on opposite sides of the transversal.

Transversal

2

1

3

4

Consecutive Interior Angles – because they lie inside the two lines and on the same side of the transversal.

5

6

Corresponding Angles – because they lie in corresponding positions of each intersection.

8

7

Alternate Interior Angles – because they lie inside the two lines andon opposite sides of the transversal.

example 2 17
Example 2.17

Determine the relationship between the given angles

  • 3 and 9
    • Alternate Interior Angles
  • 13 and 5
    • Corresponding Angles
  • 4 and 10
    • Alternate Interior Angles
  • 5 and 15
    • Alternate Exterior Angles
  • 7 and 14
    • Consecutive Interior Angles
postulate 15 corresponding angles postulate

1

2

3

4

5

6

8

7

Postulate 15:Corresponding Angles Postulate
  • If two parallel lines are cut by a transversal, then corresponding angles are congruent.
    • You must know the lines are parallel in order to assume the angles are congruent.
theorem 3 4 alternate interior angles

1

2

3

4

5

6

8

7

Theorem 3.4:Alternate Interior Angles
  • If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
    • Again, you must know that the lines are parallel.
    • If you know the two lines are parallel, then identify where the alternate interior angles are.
    • Once you identify them, they should look congruent and they are.
theorem 3 5 consecutive interior angles

1

2

3

4

5

6

8

7

Theorem 3.5:Consecutive Interior Angles
  • If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary.
    • Again be sure that the lines are parallel.
    • They don’t look to be congruent, so they MUST be supplementary.

180o

=

+

+

=

180o

theorem 3 6 alternate exterior angles

1

2

4

3

8

7

5

6

Theorem 3.6:Alternate Exterior Angles
  • If two parallel lines are cut by a transversal, then alternate exterior angles are congruent.
    • Again be sure that the lines are parallel.
example 2 18
Example 2.18

Find the missing angles for the following:

120o

120o

60o

120o

140o

140o

105o

70o

110o

105o

110o

example 2 19
Example 2.19

Solve for x

3x + 15 = 60

2x – 4 = 92

3x = 45

2x = 96

x = 15

x = 48

5x – 10 + 75 = 180

x – 10 = 100

5x + 65 = 180

x = 110

5x = 115

x = 23

parallel versus skew
Parallel versus Skew
  • Two lines are parallel if they are coplanar and do not intersect.
      • The short-hand symbol for being parallel is //.
  • Lines that are not coplanar and do not intersect are called skew lines.
    • These are lines that look like they intersect but do not lie on the same piece of paper.
  • Skew lines go in different directions while parallel lines go in the same direction.
example 2 20
Example 2.20

Complete the following statements using the words parallel, skew, perpendicular.

  • Line WZ and line XY are _________.
    • parallel
  • Line WZ and line QW are ________.
    • perpendicular
  • Line SY and line WX are _________.
    • skew
  • Plane WQR and plane SYT are _________.
    • parallel
  • Plane RQT and plane WQR are _________.
    • perpendicular
  • Line TS and line ZY are __________.
    • skew
  • Line WX and plane SYZ are __________.
    • parallel.
homework 2 5
Homework 2.5
  • Lesson 2.5
    • p11-12
  • Due Tomorrow
  • Unit 2 Test
    • Monday, October 11th
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