points, lines, planes, and angles · 2015. 2. 2. · points, lines, planes, and angles 3 postulate:...

Points, Lines, Planes, and Angles

1

Points, Lines, Planes, &

Angles

www.njctl.org

Table of ContentsPoints, Lines, & PlanesLine Segments

Distance between points

Angles & Angle RelationshipsAngle Addition Postulate

Pythagorean Theorem

Midpoint formula

Simplifying Perfect Square Radical ExpressionsRational & Irrational NumbersSimplifying NonPerfect Square Radicands

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 twodimensions. A plane has no thickness.


2

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 lowercased 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 , , or

Line a

all refer to the same line

Example

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

Answer(click)


3

Postulate: two lines intersect at exactly one point.

If two nonparallel lines intersect in a plane 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

Move

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.


4

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 onthe given line

e.

f.

g.

h.

Hint

1 is the same as .

True

False

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

Answer

2 is the same as .

True

False

Answer


5

3 Line p contains just three points.

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.

Hint

MoveAnswer

4 Points D, H, and E are collinear.

True

False

Answer

Explain your answer.5Ray LJ and ray JL are opposite rays.

Yes

No

Answer

No, Opposite Rays have same endpoint but point in opposite directions

6Which of the following are opposite rays?

A and

B and

C and

D and

Answer


6

7Name the initial point of

A JB K

C L

Answer

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

a. L, K, Jb. N, I, Mc. M, N, Kd. P, 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 noncollinear 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 noncoplanar and noncollinear.

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 noncoplanar in the diagram aboveHowever, you could draw a plane to contain any three points


7

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 B

Postulate: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 noncollinearpoints 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 8Line BC does not contain point R. Are points R, B, and C collinear?

Yes

No

Answer


8

9Plane LMN does not contain point P. Are points P, M, and N coplanar?

Yes

No

Hint:

What do we know about any three points?Answer

Move

10Plane QRS contains . Are points Q, R, S, and V coplanar? (Draw a picture)

Yes

No

Answer

11Plane JKL does not contain . Are points J, K, L, and N coplanar?

Yes

No

Answer

12 and intersect at

A Point A B Point B

C Point C

D Point D

Answer


9

13Which group of points are noncoplanar with points A, B, and F on the cube below.

A E, F, B, A

B A, C, G, E

C D, H, G, C

D F, E, G, H

Answer

14Are lines and coplanar on the cube below?

Yes

No

Answer

15Plane ABC and plane DCG intersect at _____?A C

B line DC

C Line CGD they don't intersect

Answer

B

16Planes ABC, GCD, and EGC intersect at _____?A line

B point CC point A

D line

Answer B


10

17Name another point that is in the same plane as points E, G, and H

A B

B C

C D

D F

Answer

D

18Name a point that is coplanar with points E, F, and C

A H

B B

C D

D A

Answer

C

19Intersecting lines are __________ coplanar.

A Always

B Sometimes

C Never

Answer

20Two planes ____________ intersect at exactly one point.

A Always

B Sometimes

C Never

Answer


11

21A plane can __________ be drawn so that any three points are coplanerA Always

B Sometimes

C Never

Answer

22A plane containing two points of a line __________ contains the entire line.

A Always

B Sometimes

C Never

Answer

23Four points are ____________ noncoplanar.

A Always

B Sometimes

C Never

Answer

24Two lines ________________ meet at more than one point.

A Always

B Sometimes

C Never

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

Hint

Answer


12

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.

or

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

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

coordinate

10 2 3 4 5 6 7 8 9 1012345678910

A B C D E F

coordinate

AF = |8 6| = 14Distance

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.

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.


13

Definition: CongruenceEqual 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.

=

=

Example 25Find a segment that is 4 cm long

A

B

C

D

cm

Answer


14

26Find a segment that is 3.5 cm long

A

B

C

D

cmAnswer

27Find a segment that is 2 cm long

A

B

C

D

cm

Answer

28If point F was placed at 3.5 cm on the ruler, how far from point E would it be?

A 5 cmB 4 cmC 3.5 cmD 4.5 cm

cm

Answer

Segment Addition Postulate

AC

AB BC

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.


15

ExampleThe 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

AE

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 =

K, M, and P are collinear with P between K and M. PM = 2x+4, MK = 14x56, and PK = x+17

Solve for 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 + 13

1) 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


16

29We are given the following information about the collinear points:

What is , , and ?

Answer


What is ?

Answer


What is ?

Answer


What is ?

Answer


17


What is ?

Answer


What is ?

Answer

35X, 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

Answer

36Q, 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

Answer

7x + 1 + 15x + 10 = 2x + 131

22x + 11 = 2x + 131

20x = 120

x = 6


18

37B, 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

Answer

4x + 149 + 5x = 15x + 125

9x + 149 = 15x + 125

6x = 24

x = 4


Simplifying Perfect Square Radical Expressions

Can you recall the perfect squares from 1 to 169?

12 = 82 =

22 = 92 =

32 = 102 =

42 = 112 =

52 = 122 =

62 = 132 = 202 =

72 =

Square Root Of A NumberRecall: If b2 = a, then b is a square root of a.

Example: If 42 = 16, then 4 is a square root of 16

What is a square root of 25? 64? 100?


19

Square Root Of A NumberSquare roots are written with a radical symbol

Positive square root: = 4

Negative square root: = 4

Positive & negative square roots: = 4

Negative numbers have no real square roots no real roots because there is no real number that, when squared, would equal 16.

Is there a difference between

Which expression has no real roots?

&

Evaluate the expression

?

is not real

Evaluate the expression 38 ?


20

39 = ? 40

41 = ?

A 3

B 3

C No real rootsRational & Irrational

Numbers



21

Rational & Irrational Numbers is rational because the radicand (number under the

radical) is a perfect square

If a radicand is not a perfect square, the root is said to be irrational.

Ex:

42Rational or Irrational?

A Rational B Irrational






22

Simplifying NonPerfect Square Radicands


What happens when the radicand is not a perfect square?

Rewrite the radicand as a product of its largest perfect square factor.

Simplify the square root of the perfect square.

When simplified form still contains a radical, it is said to be irrational.

Try These. Identifying the largest perfect square factor when simplifying radicals will result in the least amount of work.

Ex: Not simplified! Keep going!

Finding the largest perfect square factor results in less work:

Note that the answers are the same for both solution processes


23

45Simplify

A

B

C

D already in simplified form

46Simplify

A

B

C


47Simplify

A

B

C


48Simplify

A

B

C



24

49Simplify

A

B

C


50Simplify

A

B

C


51Which of the following does not have an irrational simplified form?

A

B

C

D

2


25

52Simplify

A

B

C

D

53Simplify

A

B C

D

54Simplify

A

B

C

D

55Simplify

A

B

C

D


26

56Simplify

A

B

C

D 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

Click to see a Visual Proof

Using the Pythagorean Theorem

c2= a2 + b25

3

a = ? 9916 = = 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.


27

Example

Answer

57What is the length of side c?

Answer

Hint:

The longest side of a triangle is called the?Move

58What is the length of side a?

Hint:

Always determine which side is the hypotenuse firstMove

Answer

59What is the length of c?B

Answer


28

60What is the length of the missing side?

Answer

61What is the length of side b?

Answer

62What is the measure of x?

8

17

x

Answer

63 Calculate the value of the missing side. Leave your answer in simplest radical form.

810

Answer


29

64 Calculate the value of the missing side. Leave your answer in simplest radical form.

18

6 Answer

65 Calculate the value of x. Leave your answer in simplest radical form.

18

12

Answer

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 66A triangle has sides 30, 40 , and 50, is it a right triangle?

Yes

No

Answer


30

67A triangle has sides 9, 12 , and 15, is it a right triangle?

Yes

No

Answer

68A triangle has sides √3, 2 , and √5, is it a right triangle?

Yes

No

Answer

Distance


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.


31

(x1, y1) (x2, y1)

(x2, y2)

The distance formulacalculates the distance using the points' 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= a2 + b2

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.

DistanceThe 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 and is given by the formula:(x1, y1) (x2, y2)

d =

Note: recall that all coordinates are (xcoordinate, ycoordinate).

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 = =

=

69Calculate the distance from Point J to Point K

A B C D

Answer

D


32

70Calculate the distance from H to K

A

B

C

D

Answer

D

71Calculate the distance from Point G to Point K

A

B

C

D

Answer

72Calculate the distance from Point I to Point H

A

B

C

D

Answer

73Calculate the distance from Point G to Point H

A

B

C

D

Answer


33

74 Calculate the distance between A(3, 7) and B(8, 2). 75 Calculate the distance between A(3, 7) and B(1, 3).

76 Calculate the distance between A(3, 0) and B(10, 6).

Midpoints



34

Midpoint of a line segmentA 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 TheoremThe midpoint of a segment joining points with coordinates and is the point with coordinates

(x1, y1)(x2, y2)

Calculating Midpoints in a Cartesian Plane

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 xcoordinate first. (x, y)

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

77Find the midpoint coordinates (x,y) of the segment connecting points A(1,2) and B(5,6)

A (4, 3) B (3, 4) C (6, 8) D (2.5, 3)

Hint:

Always label the points coordinates first

Answer


35

78Find the midpoint coordinates (x,y) of the segment connecting the points A(2,5) and B(4, 3)

A (1, 1) B (3, 8) C (8, 3) D (1, 1)

Answer

79Find the coordinates of the midpoint (x, y) of the segment with endpoints R(4, 6) and Q(2, 8)

A (1, 1) B (1, 1) C (1, 1) D (1, 1)

Answer

80Find the coordinates (x, y) of the midpoint of the segment with endpoints B(1, 3) and C(7, 9)

A (3, 3) B (6, 4) C (4, 6) D (4, 6)

Answer

81Find the midpoint (x, y) of the line segment between A(1, 3) and B(2,2)

A (3/2, 5/2) B (1/2, 5/2)

C (1/2, 3) D (3, 1/2) A

nswer


36

Example: Finding the coordinates of an endpoint of an segment

Use the midpoint formula towrite equations using x and y.

82Find the other endpoint of the segment with the endpoint (7,2) and midpoint (3,0)

A (1, 2) B (2, 1)

C (4, 2) D (2, 4)

Answer

A

83Find the other endpoint of the segment with the endpoint (1, 4) and midpoint (5, 2)

A (11, 8) B (9, 0) C (9, 8)

D (3, 1)

Answer

C


Angles&

Angle Relationships


37

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

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

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.

Angle MeasuresAngles 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

J

K

L M

N

OP

Example

Challenge Questions


38

Angle RelationshipsOnce 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

Complementary AnglesA 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

Supplementary AnglesSupplementary 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 nonshared sides form a line.

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


39

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.

Two angles are complementary. The larger angle is twice the size of the smaller angle. What is the measure of both angles?

Hint:

Choose a variable for the angle. What is a complement?

84An angle is 34° more than its complement.

What is its measure?

Move

Answer angle = (90 x) + 34

angle = complement + 34

85An angle is 14° less than its complement.

What is the angle's measure?

Hint:

What is a complement?Choose a variable for the angleMove

Answer


40

86An angle is 98 more than its supplement.

What is the measure of the angle?

Hint:

Choose a variable for the angleWhat is a supplement?

Answer

Move

87An angle is 74° less than its supplement.

What is the angle?

Answer

angle = supplement 74

88An angle is 26° more than its supplement.

What is the angle?

Answer

angle = supplement + 26

89 and are a linear pair. What is the value

of x if and .


41

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 90 Given m∠ABC = 22° and m∠DBC = 46°.

Find m∠ABD

Hint:

Always label your diagram with the information givenMove

Answer


42

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

A 28

B 15

C 11

D 11764°

53°

Answer

92 Given m∠ABD = 95° and m∠CBA = 48°.

Find m∠DBC

Answer

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

Find m∠HLJ

Answer

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

Find m∠SRT

Answer


43

95

Hint:

Draw a diagram and label it with the given information

C is in the interior of ∠TUV.

If m∠TUV = (10x + 72)⁰,

m∠TUC = (14x + 18)⁰ and

m∠CUV = (9x + 2)⁰

solve for x.

Answer

Move

96

Hint:

Draw a diagram and label it with the given informationMove

Answer

11x + 66 = 5x + 3 + 13x +7

11x + 66 = 18x + 10

7x = 56

x = 8

D is in the interior of ∠ABC.

If m∠CBA = (11x + 66)⁰,

m∠DBA = (5x + 3)⁰ and

m∠CBD= (13x + 7)⁰

solve for x.

97

Answer

F is in the interior of ∠DQP.

If m∠DQP = (3x + 44)⁰,

m∠FQP = (8x + 3)⁰ and

m∠DQF= (5x + 1)⁰

solve for x

points, lines, planes, and angles · 2015. 2. 2. · points, lines, planes, and angles 3 postulate:...

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