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New Jersey Center for Teaching and Learning

Progressive Mathematics Initiative

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Points, Lines, Planes, &

Angles

www.njctl.org

October 4, 2011




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Use Slide Show View to Administer Assessment ItemsTo administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 17 for an example.)

Setting the PowerPoint View

Table of Contents

Points, Lines, & PlanesLine Segments

Distance between points

Locus & ConstructionsAngles & Angle RelationshipsAngle Addition PostulateAngle Bisectors & Constructions

Pythagorean Theorem

Midpoint formula

Click on the topic to go to that section

Points, Lines, & Planes

Return to Table of Contents

Definitions

An "undefined term" is a word or term that does not require further explanation. There are three undefined terms in geometry:

Points - A point has no dimensions (length, width, height), it is usually represent by a capital letter and a dot on a page. It shows position only.

Lines - composed of an unlimited number of points along a straight path. A line has no width or height and extends infinitely in opposite directions.

Planes - a flat surface that extends indefinitely in two-dimensions. A plane has no thickness.

Points & LinesA television picture is composed of many dots placed closely together. But if you look very closely, you will see the spaces.

However, in geometry, a line is composed of an unlimited/infinite number of points. There are no spaces between the point that make a line. You can always find a point between any two other points.

The line above would b called line or line

A B

Points are labeled with letters. (Points A, B, or C)

Lines are named by using any two points OR by using a single lower-cased letter. Arrowheads show the line continues without end in opposite directions.

Line , , or

Line a

all refer to the same line

Collinear Points - Points D, E, and F above are called collinear points, meaning they all lie on the same line.

Points A, B, and C are NOT collinear point since they do not lie on the same (one) line.

Postulate: Any two points are always collinear.

, ,

Line a

Line , orall refer to the same line

ExampleGive six different names for the line that contains points U, V, and W.

Answer(click)

Postulate: two lines intersect at exactly one point.

If two non-parallel intersect they do so at only one point.

and intersect at K.

Examplea. Name three points that are collinearb. Name three sets of points that are noncollinearc. What is the intersection of the two lines?

a. A, D, Cb. A,B,D / A,C,B / C,D,B (others)

Answer

or

Rays are also portions of a line.

is read ray AB.

Rays start at an initial point, here endpoint A, and continues infinitely in one direction.

Ray has a different initial point, endpoint B, and continues infinitely in the direction marked.

Rays and are NOT the same. They have different initial points and extend in different directions.

Suppose point C is between points A and B

Rays and are opposite rays.

Recall: Since A, B, and C all lie on the same line, we know they are collinear points.

Similarly, segments and rays are called collinear, if they lie on the same line. Segments, rays, and lines are also called coplanar if they all lie on the same plane.

Opposite rays are two rays with a common endpoint that point in opposite directions and form a straight line.

Example

Name a point that is collinearwith the given points.

a. R and P

b. M and Q

c. S and N

d. O and P

Example

Name two opposite rays on the given line

e.

f.

h

g.

Hint

1 is the same as .

Read the notation carefully. Are they asking about lines, line segments, or rays?

True

False

2 is the same as

True

False

Remember that even though only three points are marked, a line is composed of an infinite number of points. You can always find another point in between two other points.

Answer

3 Line p contains just three points

True

False

4 Points D, H, and E are collinear.

True

False

5 Points G, D, and H are collinear.

True

False

6 Ray LJ and ray JL are opposite rays.

Yes

No

Explain your answer?

and and

and

and

A

BC

D

7 Which of the following are opposite rays?

A

B

C

8 Name the initial point of

J

KL

A

BC

9 Name the initial point of

J

KL

abcd

Are the three points collinear? If they are, name the line they lie on.

L, K, JN, I, MM, N, KP, M, I

PlanesCollinear points are points that are on the same line.

F,G, and H are three collinear points. J,G, and K are three collinear points.J,G, and H are three non-collinear points.F, G, H, and I are coplanar.

Coplanar points are points that lie on the same plane.

F,G, and H are coplanar in addition to being collinear.G, I, and K are non-coplanar and non-collinear.

F, G, H, and J are also coplanar, but the plane is not drawn.

Any three noncollinear points can name a plane.

Planes can be named by any three noncollinear points: - plane KMN, plane LKM, or plane KNL- or, by a single letter such as Plane R (all name the same plane)

Coplanar points are points that lie on the same plane:- Points K, M, and L are coplanar- Points O, K, and L are non-coplanar in the diagramHowever, you could draw a plane to contain any three points

As another example, picture the intersections of the four walls in a room with the ceiling or the floor. You can imagine a line laying along the intersections of these planes.

A BPostulate:If two planes intersect, they intersect along exactly one line.

The intersection of the two planes above is shown by line

Postulate: Through any three noncollinear points there is exactly one plane.

Name the following points:

A point not in plane HIE

A point not in plane GIE

Two points in both planes

Two points not on

Example

10 Line BC does not contain point R. Are points R, B, and C collinear?

Yes

No

Hint: what do we know about any three points?

11 Plane LMN does not contain point P. Are points P, M, and N coplanar?

Yes

No

12 Plane QRS contains . Are points Q, R, S, and V coplanar? (Draw a picture)

Yes

No

13 Plane JKL does not contain . Are points J, K, L, and N coplanar?

Yes

No

AB

C

D

D

C

A

B

14 and intersect at

AB

C

D

E, F, B, A

A, C, G, E

D, H, G, C

F, E, G, H

15 Which group of points are noncoplanar with points A,B, and F.

Answer

16 Are lines and coplanar?

Yes

No

AB

C

D

C

line DC

Line CG

they don't intersect

17 Plane ABC and plane DCG intersect at ?

Answer

AB

CD

line

point C

point A

line

18 Planes ABC, GCD, and EGC intersect at ?

Answer

AB

C

D

C

D

F

19 Name another point that is in the same plane as points E, G, and H

B

Answer

20 Name a point that is coplanar with points E, F, and C

Answer

AB

C

D

B

D

A

H

A

BC

21 Intersecting lines are __________ coplanar.

Always

Sometimes Never

22 Two planes ____________ intersect at exactly one point.

A

BC

Always

Sometimes Never

23 A plane can __________ be drawn so that any three points are coplaner

A

BC

Always

Sometimes Never

24 A plane containing two points of a line __________ contains the entire line.

A

BC

Always

Sometimes Never

25 Four points are ____________ noncoplanar.

A

BC

Always

Sometimes Never

Look what happens if I place line y directly on top of line x.

26 Two lines ________________ meet at more than one point.

Hint

A

BC

Always

Sometimes

Never

Line Segments


Line Segments

Line segments are portions of a line.or

endpoint endpoint

is read segment AB..Line Segment or are different names for the same segment.

It consists of the endpoints A and B and all the points on the line between them.

or

coordinate

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

A B C D E F

coordinate

AF = |-8 - 6| = 14

Distance

A F

Coordinates indicate the point's position on the number line.

The symbol AF stands for the length of . This distance from A to F can be found by subtracting the two coordinates and taking the absolute value.

Ruler PostulateOn a number line, every point can be paired with a number and every number can be paired with a point.

X

Why did we take the Absolute Value when calculating distance?

When you take the absolute value between two numbers, the order in which you subtract the two numbers does not matter

In our previous slide, we were seeking the distance between two points.

Distance is a physical quantity that can be measured - distances cannot be negative.

Definition: Congruence

Equal in size and shape. Two objects are congruent if they have the same dimensions and shape.

Roughly, 'congruent' means 'equal', but it has a precise meaning that you should understand completely when you consider complex shapes.

Line Segments are congruent if they have the same length. Congruent lines can be at any angle or orientation on the plane; they do not need to be parallel.

"The line segment DE iscongruent to line segment HI."

Read as:

Definition: Parallel LinesLines are parallel if they lie in the same plane, and are the same distance apart over their entire length. That is, they do not intersect.

cm

Find the measure of each segment in centimeters.

a.

b.

CE =

AB =

Example

27 Find a segment that is 4 cm long

AB

CD

BD

DE

CD

DA

cm

28 Find a segment that is 6.5 cm long

AB

C

D

BE

DE

CD

DA

cm


AB

C

D

BE

BD

DC

AC

cm

30 Find a segment that is 2 cm long

AB

C

D

CA

BD

DC

DE

cm


AB

C

D

EB

DB

DA

CE

cm

32 If point F was placed at 3.5 cm on the ruler, how from point E would it be?

AB

C

D

4 cm

3.5 cm

4.5 cm

5 cm

cm

Segment Addition PostulateIf B is between A and C, then AB + BC = AC.Or, said another way,If AB + BC = AC, then B is between A and C.

AC

AB

Simply said, if you take one part of a segment (AB), and add it to another part of the segment (BC), you get the entire segment.

The whole is equal to the sum of its parts.

BC

Example

The segment addition postulate works for three or more segments if all the segments lie on the same line (i.e. all the points are collinear).

In the diagram, AE = 27, AB = CD, DE = 5, and BC = 6

Find CD and BE

AB BC CD DE

Start by filling in the information you are given

In the diagram, AE = 27, AB = CD, DE = 5, and BC = 6

27

56|| ||

Can you finish the rest? CD = BE =

AE

ExampleK, M, and P are collinear with P between K and MPM = 2x + 4, MK = 14x - 56, and PK = x + 17Solve for x

1) Draw a diagram and insert the information given into the diagram

2) From the segment addition postulate, we know that KP + PM = MK (the parts equal the whole)

3) (x + 17) + (2x + 4) = 14x - 56 3x + 21 = 14x - 56 + 56 + 56 3x + 77 = 14x -3x - 3x 77 = 11x 7 = x

ExampleP, B, L, and M are collinear and are in the following order:a) P is between B and Mb) L is between M and P Draw a diagram and solve for x, given: ML = 3x +16, PL = 2x +11, BM = 3x +140, and PB = 3x + 131) First, arrange the points in order and draw a diagram a) BPM b) BPLM

2) Segment addition postulate gives 3x+13 + 2x+11 + 3x+16 = 3x+140

3) Combine like terms and isolate/solve for the variable x8x + 40 = 3x + 1405x + 40 = 140 5x = 100 x = 20

AE = 20BD = 6AB = BC = CD

For next group of questions (#31-36), we are given the following information about the collinear points:

Hint: always start these problems by placing the information you have into the diagram.

33 We are given the following information about the collinear points:


What is AB, BC, and CD?



What is DE?



What is CA?



What is CE?



What is DA?



What is BE?

39 X, B, and Y are collinear points, with Y between B and X. Draw a diagram and solve for x, given: BX = 6x + 151 XY = 15x - 7BY = x - 12

40 Q, X, and R are collinear points, with X between R and Q. Draw a diagram and solve for x, given: XQ = 15x + 10 RQ = 2x + 131XR = 7x +1

41 B, K, and V are collinear points, with K between V and B. Draw a diagram and solve for x, given: KB = 5x BV = 15x + 125KV = 4x +149

The Pythagorean Theorem


Pythagoras was a philosopher, theologian, scientist and mathematician born on the island of Samos in ancient Greece and lived from c. 570–c. 495 BC.

Proof

The Pythagorean Theorem

states that in a right triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides.

c2 = a2 + b2

a

bc

Proof

Pythagorean Theorem

Using the Pythagorean Theorem

c2= a2 + b25

3

a = ? -9-916 =

a

25 = a2 + 9

a2

a=4=

In the Pythagorean Theorem, c always stands for the longest side. In a right triangle, the longest side is called the hypotenuse. The hypotenuse is the side opposite the right angle.

You will use the Pythagorean Theorem often.

Example

Answer

c2= a2 + b2

42 What is the length of side c?

(The longest side of a triangle is called the ?)

Answer

Hint: Always determine which side is the hypotenuse first

43 What is the length of side a?

B44 What is the length of c?

45 What is the length of the missing side?

46 What is the length of side b?

8

17

x

47 What is the measure of x?

are three positive integers for side lengths that satisfy

( 3 , 4 , 5 ) ( 5, 12, 13) (6, 8, 10) ( 7, 24, 25)( 8, 15, 17) ( 9, 40, 41) (10, 24, 26) (11, 60, 61)(12, 35, 37) (13, 84, 85) etc.

There are many more.

a2 + b2 = c2

Remembering some of these combinations may save you some time

Pythagorean Triples

48 A triangle has sides 30, 40 , and 50, is it a right triangle?

Yes

No

49 A triangle has sides 9, 15 , and 12, is it a right triangle?

Yes

No

50 A triangle has sides √3, 2 , and √5, is it a right triangle?

Yes

No

Distance between Points


Computing the distance between two points in the plane is an application of the Pythagorean Theorem for right triangles.

Computing distances between points in the plane is equivalent to finding the length of the hypotenuse of a right triangle.

(x1, y1) (x2, y1)

(x2, y2)

The distance formulacalculates the distance using point's coordinates.

c

Relationship between the Pythagorean Theorem & Distance Formula

c

b

a

The Pythagorean Theorem states a relationship among the sides of a right triangle.

c2= a

2 + b

2

The Pythagorean Theorem is true for all right triangles. If we know the lengths of two sides of a right triangle then we know the length of the third side.

Distance

The distance between two points, whether on a line or in a coordinate plane, is computed using the distance formula.

The Distance FormulaThe distance 'd' between any two points with coordinates (x1 ,y1) and (x2 ,y2) is given by the formula:

d =

Note: recall that all coordinates are (x-coordinate, y-coordinate).

Example

Calculate the distance from Point K to Point I

(x1, y1) (x2, y2)

d = Plug the coordinates into the distance formula

Label the points - it does not matterwhich one you label point 1 and point 2. Your answer will be the same.

KI =

KI = =

=

AB

C

D

51 Calculate the distance from Point J to Point K

AB

C

D

52 Calculate the distance from H to K

AB

CD

53 Calculate the distance from Point G to Point K

AB

C

D

54 Calculate the distance from Point I to Point H

AB

C

D

55 Calculate the distance from Point G to Point H

Midpoint Formula


Midpoint of a line segment

A number line can help you find the midpoint of a segment.

Take the coordinates of the endpoint G and H, add them together, and divide by two.

= = -1

The midpoint of GH, marked by point M, is -1.

Here's how you calculate it using the endpoint coordinates.

Midpoint Formula Theorem

The midpoint of a segment joining points with coordinates and is the point with coordinates

(x1, y1)(x2, y2)

Segment PQ contain thepoints (2, 4) and (10, 6).The midpoint M of isthe point halfway betweenP and Q.

Just as before, we find the average of the coordinates.

( , )

Remember that points are written with the x-coordinate first. (x, y)

The coordinates of M, the midpoint of PQ, are (6, 5)

Calculating Midpoints in a Cartesian Plane

Hint: Always label the points coordinates first

56 Find the midpoint coordinates (x,y) of the segment connecting points A(1,2) and B(5,6)

AB

C

D

(3, 4)

(6, 8)

(2.5, 3)

(4, 3)

57 Find the midpoint coordinates (x,y) of the segment connecting the points A(-2,5) and B(4, -3)

AB

C

D

(-3, -8)

(-8, -3)

(1, 1)

(-1, -1)

58 Find the coordinates of the midpoint (x, y) of the segment with endpoints R(-4, 6) and Q(2, -8)

AB

C

D

(1, 1)

(-1, -1)

(1, -1)

(-1, 1)

59 Find the coordinates (x, y) of the midpoint of the segment with endpoints B(-1, 3) and C(-7, 9)

AB

C

D

(6, -4)

(-4, 6)

(4, 6)

(-3, 3)

60 Find the midpoint (x, y) of the line segment between A(-1, 3) and B(2,2)

AB

C

D

(1/2, 5/2)

(1/2, 3)

(3, 1/2)

(3/2, 5/2)

Example: Finding the coordinates of an endpoint of an segment

Use the midpoint formula towrite equations using x and y.

61 Find the other endpoint of the segment with the endpoint (7,2) and midpoint (3,0)

AB

C

D

(-2, -1)

(4, 2)

(2, 4)

(-1, -2)

62 Find the other endpoint of the segment with the endpoint (1, 4) and (5, -2)

AB

C

D

(3, 1)

(3, -3)

(-3, 3)

(1, 3)

Locus&

Constructions


Introduction to Locus

Definition:Locus is a set of points that all satisfy a certain condition, in this course, it is the set of points that are the same distance from something else.

All points equidistant from a given line is a parallel line.Theorem: locus from a given line

Theorem: locus between two pointsAll points on the perpendicular bisector of a line segment connecting two points are equidistant from the two points.

Theorem: locus between two linesThe locus of points equidistant from two given parallel lines is a parallel line midway between them.

locus: equidistant from two points

The locus of points equidistant from two points, A and B, is the perpendicular bisector of the line segment determined by the two points.

The distance (d) from point A to the locus is equal to the distance (d') from Point B to the locus. The set of all these points forms the red line and is named the locus.

X

Y

Point M is the midpoint of . X is equidistant (d=d') from A and B. Y lies on the locus; it is also equidistant from A and B.

Dividing a line segment into x congruent segments. Let us divide AB into 3 equal segments - we could choose any number of segments.

3. Set the compass width to CB.

2. Set the compass on A, and set its width to a bit less than 1/3 of the length of the new line.

Step the compass along the line, marking off 3 arcs. Label the last C.

1. From point A, draw a line segment at an angle to the given line, and about the same length. The exact length is not important.

Constructions

4. Using the compass set to CB, draw an arc below A

5. With the compass width set to AC, draw an arc from B intersecting the arc you just drew in step 4. Label this D.

6. Draw a line connecting B with D

7. Set the compass width back to AC and step along DB making 4 new arcs across the line

8. Draw lines connecting the arc along AC and BD. These lines intersect AB and divide it into 3 congruent segments.

63 Point C is on the locus between point A and point B

True

False

64 Point C is on the locus between point A and point B

True

False

65 How many points are equidistant from the endpoints of ?

AB

C

D

1

0

infinite

2

66 You can find the midpoint of a line segment by

AB

C

D

constructing the midpoint

finding the intersection of the locus and line segment

all of the above

measuring with a ruler

AB

C

D

the midpoint of a segment

the set of all points equidistant from two other points

a set of points

a straight line between two points

67 The definition of locus

Divide the line segment into 3 congruent segments.

Example: Construction

Angles&

Angle RelationshipsReturn to Table of Contents

AB

(Side)

(Side)

32°

C

(Vertex)

The measure of the angle is 32 degrees.

"The measure of is equal to the measure of ..."

The angle shown can be called , , or .

When there is no chanceof confusion, the angle may also be identified by its vertex B.

The sides of are CB and AB

An angle is formed by two rays with a common endpoint (vertex)

Identifying Angles

Two angles that have the same measure are congruent angles.

Interior

ExteriorThe single mark through the arc shows that the angle measures are equal

We read this as is congruent to

The area between the rays that form an angle is called the interior. The exterior is the area outside the angle.

Angles are measured in degrees, using a protractor.Every angle has a measure from 0 to 180 degrees.Angles can be drawn any size, the measure would still be the same.

A

B C

D

The measure ofis 23° degrees

is a 23° degree angleThe measure ofis 119° degrees

is a 119° degree angle

In and , notice that the vertex is written in between the sides

Angle Measures

J

K

L M

N

OP

Challenge Questions

X

Example

Once we know the measurements of angles, we can categorize them into several groups of angles:

right = 90°

Two lines or line segments that meetat a right angle are said to be perpendicular.

straight = 180°180°

0° < acute < 90° 90° < obtuse < 180°

180° < reflex angle < 360°

Link

Angle Relationships

A pair of angles are called complementary angles if the sum of their degree measurements equals 90 degrees. One of the angles is said to be the complement of the other.

These two angles are complementary (58° + 32° = 90°)

We can rearrange the angles so they are adjacent, i.e. share a common side and a vertex. Complementary angles do not have to be adjacent. If two adjacent angles are complementary, they form a right angle

=

Complementary Angles

+

Supplementary Angles

Supplementary angles are pairs of angles whose measurements sum to 180 degrees. Supplementary angles do not have to be adjacent or on the same line; they can be separated in space. One angle is said to be the supplement of the other.

Definition: Adjacent Angles

are angles that have a common ray coming out of the vertex going between two other rays. In other words, they are angles that are side by side, or adjacent.

If the two supplementary angles are adjacent, having a common vertex and sharing one side, their non-shared sides form a line.

A linear pair of angles are two adjacent angles whose non-common sides on the same line. A line could also be called a straight angle with 180°

=+

Example

Solution:Choose a variable for the angle - I'll choose "x"

Example

Let x = the angle

90 = 2x + x90 = 3x30 = x

Since the angles are complementary we know their summust equal 90 degrees.

Hint: Choose a variable for the angle What is a complement?

angle = (90 - x) + 34x = 90 - x +34

2x = 124x = 62

angle = complement + 34

Answer

68 An angle is 34° more than its complement.

What is its measure?

angle = (90 - x) - 14x = 90 - x - 14

2x = 90 - 142x = 76

x = 38

angle = complement - 14

Hint: What is a complement? Choose a variable for the angle

69 An angle is 14° less than its complement.

What is the angle's measure?

Answer

angle = (180 - x) + 98x = 180 - x + 98

2x = 278x = 139

Hint: Choose a variable for the angle What is a complement?

70 An angle is 98 more than its supplement.

What is the measure of the angle?

o

Answer

angle = supplement - 74x = (180 - x) - 74

2x = 180 - 742x = 106

x = 53

71 An angle is 74° less than its supplement.

What is the angle?

Answer

angle = supplement + 26x = (180 - x) + 26

2x = 180 + 262x = 206

x = 106

72 An angle is 26° more than its supplement.

What is the angle?

Answer

Angle Addition Postulate


Angle Addition Postulateif a point S lies in the interior of PQR, ∠then PQS + SQR = PQR. ∠ ∠ ∠

+m PQS = 32°∠ m SQR = 26°∠ m PQR = 58°∠

58°32°

26°

Just as from the Segment Addition Postulate,"The whole is the sum of the parts"

Example

Hint: Always label your diagram with the information given

73 Given m ABC = 23° and m DBC = 46°. ∠ ∠

Find m ABD∠

74 Given m OLM = 64°∠ and m OLN = 53°. Find ∠m NLM∠

AB

C

D

15

11

117

28

75 Given m ABD = 95° and m CBA = 48°. ∠ ∠

Find m DBC∠

76 Given m KLJ = 145° and m KLH = 61°. ∠ ∠

Find m HLJ ∠

77 Given m TRQ = 61° and m SRQ = 153°. ∠ ∠

Find m SRT∠

Hint: Draw a diagram and label it with the given information

78

79


80


Angle Bisectors & Constructions


Angle Bisector: A ray or line which starts at the vertex and cuts an angle into two equal halves

bisects

Bisect means to cut it into two equal parts. The 'bisector' is the thing doing the cutting.

The angle bisector is equidistant from the sides of the angle when measured along a segment perpendicular to the sides of the angle.

Definition: Angle Bisector

Try this!

Bisect the angle

Try this!

Bisect the angle

Documents

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