Geometry Points, Lines, Planes, & Angles

Geometry

Points, Lines, Planes, & Angles

Table of Contents

Points, Lines, & PlanesLine Segments

Distance between points

Locus & Constructions

Angles & Angle Addition Postulate

Angle Pair Relationships

Angle Bisectors & Constructions

Pythagorean Theorem

Midpoint formula

Area of Figures in Coordinate Plane

Table of Contents for VideosDemonstrating Constructions

Congruent Segments

Midpoints

Equilateral Triangles

Circle

Angle Bisectors

Congruent Angles

Points, Lines, & Planes

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 & Lines

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

....................................

A B

You can always find a point between any two other The line above would be called line or line

However, in geometry, a line is composed of an unlimited (infinite) number of points. There are no spaces between the points that make a line.

Line , , or

Line a

all refer to the same linePoints are labeled with letters.

Lines are named by using any two points OR by using a single lower-case letter. Arrowheads show that the line continues

without end in opposite directions.

Points & Lines

Postulate: Any two points are always collinear.

Line , , or

Line a

all refer to the same line

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

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

Collinear Points

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

Answ

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Postulate: two lines intersect at exactly one point.

If two non-parallel lines intersect in a plane they do so at only one point. and intersect at K.

Intersecting Lines

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

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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.

Rays

or

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

Naming Rays

Suppose point C is between points A and B

Rays and are opposite rays.

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

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.

Collinear Rays

ExampleName a point that is collinear with the given points.

a. R and P

b. M and Q

c. S and N

d. O and P

ExampleName two opposite rays on the 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?

click

Answ

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2 is the same as .

True

False

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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

click to reveal

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4 Points D, H, and E are collinear.

True

False

Answ

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5 Points G, D, and H are collinear.

True

False

Answ

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Explain your answer.

6 Ray LJ and ray JL are opposite rays.

Yes

No

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7 Which of the following are opposite rays?

A and

B and

C and

D and Answ

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8 Name the initial point of

A JB K

C L

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9 Name the initial point of

A J

B K

C L

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ExampleAre 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

Points K, M, and L are coplanar.

Points O, K, and L are non-coplanar in the diagram above.

However, you could draw a plane to contain any three points

Planes

Plane KMN, plane LKM, or plane KNL or, by a single letter such as Plane R. (These all name the same plane)

Coplanar points are points that lie on the same plane:

Planes can be named by any three noncollinear points:

Collinear points are points that are on the same line.

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

Coplanar points are points that lie on the same plane.

Any three non-collinear points can name a plane.

F, G, H, and I are coplanar.F, G, H, and J are also coplanar, but the plane is not drawn.F,G, and H are coplanar in addition to being collinear.G, I, and K are non-coplanar and non-collinear.

A B

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.

If two planes intersect, they intersect along exactly one line.

The intersection of these two planes is shown by line

Intersecting Planes

Through any three non-collinear 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

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

Yes

No

Hint:

What do we know about any three points?click to reveal

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

Yes

No

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

Yes

No

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14 and intersect at

A Point A

B Point B

C Point C

D Point D Answ

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15 Which 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

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16 Are lines and coplanar on the cube below?

Yes

No

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17 Plane ABC and plane DCG intersect at _____?

A C

B line DC

C Line CG

D they don't intersect Answ

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18 Planes ABC, GCD, and EGC intersect at _____?

A line

B point C

C point A

D line

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

A B

B C

C D

D F Answ

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

A H

B B

C D

D A

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21 Intersecting lines are __________ coplanar.

A Always

B Sometimes

C Never

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22 Two planes ____________ intersect at exactly one point.

A Always

B Sometimes

C Never

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

A Always

B Sometimes

C Never

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

A Always

B Sometimes

C Never

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25 Four points are ____________ noncoplanar.

A Always

B Sometimes

C Never

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26 Two 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.

click to revealHINT

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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

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

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.

Ruler Postulate

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.

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| = 14Distance

A F

 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.

Definition: Congruence

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.

Read as:"The line segment DE is congruent to line segment HI."

Congruent Segments

Constructing Congruent SegmentsGiven:

1. Draw a reference line w/ your straight edge. Place a reference point to indicate where your new segment start on the line.

2. Place your compass point on point A.

3. Stretch the compass out so that the pencil tip is on point B.

Given: AB

Constructing Congruent Segments(cont'd)

4. Without stretching out your compass, place the compass point on the reference point & swing the pencil so that it crosses the reference line.

5. Make a point where the pencil crossed your line. Label your new segment.

Constructing Congruent Segments(cont'd)

Constructing Congruent Segments

Try This! Construct a Congruent Segment on the horizontal line.

1)

Teac

her

Not

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Constructing Congruent Segments

Try This! Construct a Congruent Segment on the slanted line.

2)

Video Demonstrating Constructing Congruent Segments using Dynamic Geometric Software

Click here to see video

http://youtu.be/E3dabgYwzZI

http://youtu.be/E3dabgYwzZI

Definition: Parallel Lines

Lines 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

8 - 2 = 6 cm

1.5 cm

click

click

27 Find a segment that is 4 cm long

A

B

C

D

cm

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28 Find a segment that is 6.5 cm long

A

B

C

D

cm

Answ

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A

B

C

D

cm

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30 Find a segment that is 2 cm long

A

B

C

D

cm

Answ

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A

B

C

D

cm

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

A 5 cm

B 4 cm

C 3.5 cm

D 4.5 cm

cm

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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.

If B is between A and C, Then AB + BC = AC.

Or, If AB + BC = AC, then B is between A and C.

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

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?

x x

2x+11 = 272x = 16x = 8 = CD

= 19

click

click

click

click

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

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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 x

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For next six questions 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 , , and ?

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What is ?

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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

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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

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7x + 1 + 15x + 10 = 2x + 13122x + 11 = 2x + 131

20x = 120x = 6

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

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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.

The theorem 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

Pythagorean Theorem

Pythagorean Theory Visual Proof 1

Pythagorean Theory Visual Proof 2

Using the Pythagorean Theorem

c2= a2 + b25

3

a = ?-9-9

16 = = 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

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42 What is the length of side c?

The longest side of a triangle is called the ___________

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43 What is the length of side a?

Hint:

Always determine which side is the hypotenuse firstclick to reveal

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44 What is the length of c?B

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45 What is the length of the missing side?

Answ

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46 What is the length of side b?

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47 What is the measure of x?

8

17

x

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are three positive integers for side lengths that satisfy a2 + b2 = c2

( 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.  

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

Answ

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

Yes

No

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

Yes

No

Answ

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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.

Computing distance

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.

(x1, y1)

c

c

b

a

c2= a2 + b2

(x2, y1)

(x2, y2)

The distance formulacalculates the distance using the points' coordinates.

Relationship between the Pythagorean Theorem & Distance Formula

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

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

The Distance Formula

The 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 (x-coordinate, y-coordinate).

d =

KI =

Example

Calculate the distance from Point K to Point I

(x1, y1)

Plug the coordinates into the distance formula

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

(x2, y2)

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51 Calculate the distance from Point J to Point K

A

B

C

D

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52 Calculate the distance from H to K

A

B

C

D

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53 Calculate the distance from Point G to Point K

A

B

C

D

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54 Calculate the distance from Point I to Point H

A

B

C

D

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55 Calculate the distance from Point G to Point H

A

B

C

D

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Area of Figures in the Coordinate Plane

Area Formulas:Area of a Triangle: A = bh

Area of a Rectangle: A = lw

1 2

Calculating Area of Figures in the Coordinate Plane

Steps to calculate the area:1) Calculate the desired distances using the distance formula

> Ex: base & height in a triangle> Ex: length & width in a rectangle

2) Calculate the area of the shape

Example: Calculate the area of the rectangle.


> = FG & w = EF


=

= w

Example: Calculate the area of the triangle.


> b = AC & h = BD Answ

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Example: Calculate the area of the triangle.


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56 Calculate the area of the rectangle.

A

B

C 10

D 20 Answ

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57 Calculate the area of the triangle.

A 75

B 144

C 84.5

D 169

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58 Calculate the area of the rectangle.

A 10,000

B 100

C 50

D

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59 Calculate the area of the triangle.

A

B

C 22.5

D 45

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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

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

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

Midpoint Formula Theorem

The midpoint of a segment joining points with coordinates and is the point with coordinates(x2, y2)

(x1, y1)

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 x-coordinate first. (x, y)

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

60 Find 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

click

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61 Find 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)

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62 Find 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)

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63 Find 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)

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64 Find 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) Answ

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Example: Finding the coordinates of an endpoint of an segment

Use the midpoint formula towrite equations using x and y.

65 Find 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)

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66 Find 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)

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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 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.

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

We can find the midpoint of a line segment by constructing the locus between two points.

Given

Bisect a Line Segment


Given

1. Place the point of the compass on A and open it so the pencil is more than halfway, it does not matter how far, to point B



Given

2. Keeping the compass locked, place the point of the compass on B and draw an arc so that it intersects the one you just drew



Given

3. Draw a line through the arc intersections. M marks the midpoint of

M



Given

2. Keeping the compass locked, place the point of the compass on B and draw an arc so that it intersects the one you just drew


M

1. Place the point of the compass on A and open it so the pencil is more than halfway, it does not matter how far, to point B



Try this!Construct the midpoint of the segment below.

3)



4)

We can also construct the locus between two points with string, a rod (e.g. marker, closed pen, etc) & a pencil.

Given

1. Place the rod on A and extend your piece of string out. The pencil should be more than halfway across the segment. It does not matter how far, to point B.

Bisect a Line Segment w/ a string & rod

Given

Bisect a Line Segment w/ a string & a rodWe can also construct the locus between two points with string, a rod (e.g. marker, closed pen, etc) & a pencil.

2. Place the rod on B and extend your piece of string out. The pencil should be more than half way across the segment. It does not matter how far, to point A.

Given


M

We can also construct the locus between two points with string, a rod (e.g. marker, closed pen, etc) & a pencil.

Bisect a Line Segment w/ a string & a rod


Given


M

1. Place the rod on A and extend your piece of string out. The pencil should be more than halfway across the segment. It does not matter how far, to point B.

2. Place the rod on B and extend your piece of string out. The pencil should be more than half way across the segment. It does not matter how far, to point A.




5)

Bisect a Line Segment w/ a string & a rodTry this!Construct the midpoint of the segment below.

Bisect a Line Segment w/ Paper Folding1. On patty paper, draw a segment. Label the endpoints D & E.

2. Fold your patty paper so that point D lines up with point E. Crease the fold.

3. Unfold your patty paper. Draw a line along the fold. 4. Draw & label your

midpoint M.

FOLD

Othe

r pap

er e

dge

Bisect a Line Segment w/ Paper FoldingTry this!Construct the midpoint of the segment below.

7)

Bisect a Line Segment w/ Paper FoldingTry this!Construct the midpoint of the segment below.

8)

Constructions

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.

C

D

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 3 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.

Divide the line segment into 3 congruent segments.

Example: Construction

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

True

False

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

True

False

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69 How many points are equidistant from the endpoints of ?

A 2

B 1

C 0

D infinite

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70 You can find the midpoint of a line segment by

A measuring with a ruler

B constructing the midpoint

C finding the intersection of the locus and line segment

D all of the above

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71 The definition of locus

A a straight line between two points

B the midpoint of a segment

C the set of all points equidistant from two other points

D a set of points

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Videos Demonstrating Constructing Midpoints using Dynamic Geometric Software

Click here to see video using compass and segment tool

Click here to see video using menu options

http://youtu.be/5C6BOcgclnE

http://youtu.be/5C6BOcgclnE

http://youtu.be/F9SXwe26RrQ

http://youtu.be/F9SXwe26RrQ

What is a circle?

Construction of a Circle

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Extension: Construction of a Circle

1. Find the midpoint of a line segment 1st, using any of the methods that we discussed.

M

Teac

her

Not

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Extension: Construction of a Circle2. Take your compass (or rod, string & pencil) and place the tip (or rod) on your Midpoint. Extend out your pencil as much as you would like. Turn your compass (or string & pencil) 3600 to make your circle.

M


Try this!Create a circle using the segment below.

9)


Try this!Create a circle using the segment below.

10)

Video Demonstrating Constructing a Circle using Dynamic Geometric Software


http://youtu.be/kUAdEA_C8_s

http://youtu.be/kUAdEA_C8_s

What is an equilateral triangle?

Construction of an Equilateral Triangle

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Extension: Construction of an Equilateral Triangle1. Construct a segment of any length.

2. Take your compass and place the tip on one endpoint & make an arc.

3. Take your compass and place the tip on the other endpoint & make an arc that intersects your 1st arc. 4. Make a point where your two

arcs intersect. Connect your 3 points.

C


Try this!Create an equilateral triangle using the segment below.

11)



12)

Extension: Construction of an Equilateral Triangle w/ rod, string & pencil1. Construct a segment of any length.

2. Place your rod on one endpoint & the pencil on the other. Extend your pencil to make an arc.

3. Switch the places of your rod & pencil tip. Make an arc that intersects your 1st arc. 4. Make a point where your two

arcs intersect. Connect your 3 points. C

Construction of an Equilateral Triangle w/ rod, string & pencil


13)

Construction of an Equilateral Triangle w/ rod, string & pencil


14)

Video Demonstrating Constructing Equilateral Triangles using Dynamic Geometric Software


http://youtu.be/2Y2p6xL9220

http://youtu.be/2Y2p6xL9220

Angles & Angle Addition

Postulate

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

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

The sides of are rays BC and BA

Identifying Angles

Two angles that have the same measure are congruent angles.

Interior

Exterior

The 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.

Congruent angles

Constructing Congruent angles

1. Draw a reference line w/ your straight edge. Place a reference point to indicate where your new segment starts on the line.

2. Place your compass point on the vertex (point G).

Given: <FGH

4. Swing an arc with the pencil so that it crosses both sides of <FGH.

3. Stretch the compass to any length so long as it stays ON the angle.

Constructing Congruent angles (cont'd)5. Without changing the span of the compass, place the compass tip on your reference point & swing an arc that goes through the line & extends above the line.

6. Go back to <FGH & measure the width of the arc from where it crosses one side of the angle to where it crosses the other side of the angle.

The width of the arc is 2.2 cm

Constructing Congruent angles (cont'd)7. With this width, place the compass point on the reference line where your new arc crosses the reference line. Mark off this width on your new arc.

8. Connect this new intersection point to the starting point on the reference line.

The width of the arc is 2.2 cm


Try this! Construct a congruent angle on the given line.

15)


Try this! Construct a congruent angle on the given line.

16)

Video Demonstrating Constructing Congruent Angles using Dynamic Geometric Software


http://youtu.be/0K65Qop1MUo

http://youtu.be/0K65Qop1MUo

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 angle

The 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

32o

63o

90o

135o

180o

Example

click

click clic

k

click

click

J

K

L M

N

OP

Example

Challenge Questions

148-117 31o

117- 90 27o

90-45 45o

45o

click

click

click

click

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

right = 90° straight = 180°180°

0° < acute < 90° 90° < obtuse < 180°

180° < reflex angle < 360°

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

Click here for a Math Is Fun website

activity.

http://www.mathsisfun.com/geometry/images/angles-drag.swf

http://www.mathsisfun.com/geometry/images/angles-drag.swf

Adjacent Angles

Adjacent angles are angles that have a common ray coming out of the vertex going between two other rays.

J

KO

P

In other words, they are angles that are side by side, or adjacent.

Angle Addition Postulate

if 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

72 Given m∠ABC = 22° and m∠DBC = 46°.

Find m∠ABD

Hint:

Always label your diagram with the information given

click

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

A 28

B 15

C 11

D 11764°

53°

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74 Given m∠ABD = 95° and m∠CBA = 48°.

Find m∠DBC

Answ

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

Find m∠HLJ

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

Find m∠SRT

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77

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 andm∠CUV = (9x + 2)⁰solve for x.

click

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78 D is in the interior of ∠ABC.If m∠CBA = (11x + 66)⁰,m∠DBA = (5x + 3)⁰ andm∠CBD= (13x + 7)⁰solve for x.

Hint:

Draw a diagram and label it with the given informationclick

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79 F is in the interior of ∠DQP.If m∠DQP = (3x + 44)⁰,m∠FQP = (8x + 3)⁰ andm∠DQF= (5x + 1)⁰solve for x

Answ

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Angle Pair Relationships

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 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.

J

K

OP

44 degrees 136 degrees

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?

80 An angle is 34° more than its complement.

What is its measure?

click

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81 An angle is 14° less than its complement.

What is the angle's measure?

Hint:

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

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82 An angle is 98 more than its supplement.

What is the measure of the angle?

Hint:

Choose a variable for the angleWhat is a supplement?click

Answ

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83 An angle is 74° less than its supplement.

What is the angle?

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84 An angle is 26° more than its supplement.

What is the angle?

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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°

Linear Pair of Angles

Given: m∠ABC = 55oFind the measures of the remaining angles.

Vertical Angles

A

B C

D

E55o

Vertical Angles: Two angles whose sides form two pairs of opposite rays- In diagram below, ∠ABC & ∠DBE are vertical angles, and ∠ABE & ∠CBD are vertical angles.

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Given: m∠ABC = 55oWhat do you notice about your vertical angles?

Vertical Angles

C

A

B

D

E55o55

o 125o

125o

Vertical Angles Theorem: All vertical angles are congruent.click to reveal theorem

ExampleFind the m < 1, m < 2 & m < 3. Explain your answer.

36o

123

m < 2 = 36o; Vertical angles are congruent (original angle & < 2)m < 3 = 144o; Vertical angles are congruent (< 1 & < 3)

36 + m < 1 = 180-36 -36

m < 1 = 144oLinear pair angles are supplementary

85 What is the m < 1?

A 77o

B 103o

C 113o

D none of the above

77o

12 3

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A 77o

B 103o

C 113o

D none of the above

77o

12 3

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A 112o

B 78o

C 102o

D none of the above

112o

46 5

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A 112o

B 68o

C 102o

D none of the above

112o

46 5

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A 102o

B 78o

C 112o

D none of the above

112o

46 5

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ExampleFind the value of x.

(13x + 16)o(14x +

7)o

The angles shown are vertical angles, so they are congruent

13x + 16 = 14x + 7-13x -13x 16 = x + 7 - 7 - 7 9 = x

ExampleFind the value of x.

(3x + 17)o

(2x + 8)o

The angles shown are a linear pair, so they are supplementary

2x + 8 + 3x + 17 = 180 5x + 25 = 180

- 25 - 25 5x = 155 5

5 x = 31

91 Find the value of x.

A 95

B 50

C 45

D 40

(2x - 5)o85o

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A 75

B 17

C 13

D 12

(6x + 3)o75o

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A 13.1

B 14

C 15

D 122

(9x - 4)o122o

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A 12

B 13

C 42

D 138

(7x + 54)o 42o

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Angle Bisectors & Constructions

Angle Bisector

An angle bisector is 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.

Constructing Angle Bisectors1. With the compass point onthe vertex, draw an arc across each side

2. Without changing the compass setting, place the compass point on the arc intersections of the sides

X

m∠UVX = m∠WVX∠UVX ∠WVX

4. Label your point3. With a straightedge, connect the vertex to the arc intersections

Try this!

Bisect the angle

17)

Try this!

Bisect the angle18)

Constructing Angle Bisectors w/ string, rod, pencil & straightedge

1. With the rod on the vertex, draw an arc across each side.

2. Place the rod on the arc intersections of the sides & draw 2 arcs, one from each side showing an intersection point.

X

m∠UVX = m∠WVX∠UVX ∠WVX

4. Label your point3. With a straightedge, connect the vertex to the arc intersections

Try this!

Bisect the angle with string, rod, pencil & straightedge.

19)

Try this!

Bisect the angle with string, rod, pencil & straightedge.

20)

Constructing Angle Bisectors by Folding1. On patty paper, create any angle of your choice. Make it appear large on your patty paper. Label the points A, B & C

2. Fold your patty paper so that BA lines up with BC. Crease the fold.

3. Unfold your patty paper. Draw a ray along the fold, starting at point B.

4. Draw & label your point.

Try this!

Bisect the angle with folding.21)

Try this!

Bisect the angle with folding.

22)

Videos Demonstrating Constructing Angle Bisectors using Dynamic Geometric Software

Click here to see video using a compass and

segment tool

Click here to see video using the menu options

http://youtu.be/EA-NkCJtwN4

http://youtu.be/EA-NkCJtwN4

http://youtu.be/iiMgioj1zeU

http://youtu.be/iiMgioj1zeU

A

B C

D

Finding the missing measurement.

Example: ABC is bisected by BD. Find the measures of the missing angles.

52o

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H

F G

E

Example: EFG is bisected by FH. The m EFG = 56o. Find the measures of the missing angles.

56o

Answ

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Example: MO bisects LMN. Find the value of x.

L

M

N

(3x - 20)o(x + 10)o

O

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95 JK bisects HJL. Given that m HJL = 46o, what is m HJK?

Hint:

What does bisect mean?Draw & label a picture.

click to reveal

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96 NP bisects MNO Given that m MNP = 57o, what is m MNO?

Hint:

What does bisect mean?Draw & label a picture.click to reveal

Answ

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97 RT bisects QRS Given that m QRT = 78o, what is m QRS?

Answ

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98 VY bisects UVW. Given that m UVW = 165o, what is m UVY?

Answ

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D

B

A

(11x - 25)o

(7x + 3)o

C

99 BD bisects ABC. Find the value of x.

Answ

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H

F

E

(3x + 49)o

(9x - 17)o

G

100 FH bisects EFG. Find the value of x.

Answ

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I

J

L

(12x - 19)o

(7x + 1)o

K

101 JL bisects IJK. Find the value of x.

Answ

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Geometry Points, Lines, Planes, & Angles

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Transcript of Geometry Points, Lines, Planes, & Angles