geometry parallel and perpendicular · pdf fileparallel and perpendicular lines geometry ......

1

www.njctl.org

Parallel and Perpendicular Lines

Geometry

2013-08-08

Table of Contents

Angles and Parallel Lines

Slopes of Parallel Lines

Slopes of Perpendicular Lines

Proofs Involving Parallel and Perpendicular Lines

Constructing Parallel Lines

Constructing Perpendicular Lines

Videos-Table of Contents

Parallel Lines - using Menu Options

Perpendicular Lines - with Compass and Straight Edge

Perpendicular Lines - using Menu Options

2

Angles & Parallel Lines

Angles and Parallel Lines

of Contents

Jun 29-11:48 AM

Parallel, Perpendicular and Skew

How many different ways can you draw 2 lines in a plane?

Click Click


never meet are called parallel lines point are called

intersecting lines


3

Jun 29-12:19 PM

Term Definition Diagram Symbol

Parallel lines

2 lines that

lie in the

same plane

and never

intersect

k || m

Intersecting

lines

2 lines that

lie in the

same plane

and meet in

one point

no symbol

Perpendicular

lines

2 lines that

lie in the

same plane

and meet at

4 right

angles

ab

a b


Jun 29-12:08 PM

Using the following diagram, name a line that does not lie in

the same plane with HG and does not intersect HG.


Jul 3-3:08 PM


Lines that do not intersect and do not lie in the same plane are

called skew.

HG is skew with EA

HG is skew with BF

HG is skew with AB

HG is skew with AC

4

Jul 17-11:23 AM

1 Are lines a and b skew?

Yes

No

Jul 3-3:17 PM

If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.

k

P There is only 1 line through

point P parallel to line k.

Jul 3-3:24 PM

If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line.

k

P There is only 1 line through

point P perpendicular to line k.

5

Jul 3-3:25 PM

How many lines can be drawn to fit the description?

through C parallel to AB?

through C perpendicular to AB?

B

AC


2 Name all lines parallel to EF.


3

AB

6


4


5


6

Two skew lines are __________ parallel.

7


: A line that intersects two or more coplanar

lines at different points.


When a transversal intersects two lines, eight angles


2. Name the interior angles.

8

Nov 27-3:22 PM

Nov 27-3:22 PM

Nov 27-3:22 PM

9

Nov 27-3:22 PM



10

Jun 20-3:16 PM

Jun 30-10:27 AM

7 <3 and < 6 are _____.

A Corresponding Angles

B Alternate Exterior AnglesC Same-Side Exterior AnglesD Vertical Angles

Jun 30-10:41 AM

8 <1 and <3 are _____.

A Alternate Interior Angles

B Corresponding AnglesC Linear Pair of AnglesD Same-Side Interior Angles

A

CD

AB || CD and AD || BC

1 2

3 4

B

11

Jun 30-10:44 AM

9 <3 and <4 are _____.


B Same-Side Interior AnglesC Alternate Interior AnglesD Same-Side Exterior Angles

A

CD


1 2

3 4

B

Jun 30-10:56 AM

10 <3 and <6 are _____.


B Same-Side Exterior AnglesC Alternate Interior AnglesD Alternate Exterior Angles

A

CD


1

4

B

23

56

Jun 30-11:12 AM

11 <2 and <6 are ____.

A Corresponding

B Alternate Exterior AnglesC Vertical AnglesD Same-Side Exterior Angles

E None of the above

1

23

4 56

78

9

1011

12

12

Jul 6-12:08 PM

12 Name all angles corresponding with <4.

A <7

B <10C <8D <9

E Both A and B

1

23

4 56

78

9

1011

12

Lab 1

There are several theorems and postulates related to parallel lines. At this time, please go to the lab titled, "Properties of Parallel Lines".

Click here to go to the lab titled, "Propertiesof Parallel Lines"


If two parallel lines are cut by a transversal, then the corresponding angles are congruent.

According to the Corresponding Angles Postulate what angles are congruent?

13



If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.

According to the Alternate Interior Angles Theorem what angles are congruent?


14


If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.

According to the Alternate Exterior Angles Theorem what angles are congruent?



If two parallel lines are cut by a transversal, then the same-side angles are supplementary.

According to the Same-Side Angles Theorem which pairs of angles are supplementary?

15


Jun 30-4:45 PM

1. Name all of the angles congruent to <1.

2. Name all of the angles supplementary to <1.

Jun 24-7:29 PM

13 Find all of the angles congruent to <5.

A <1

B <4

C <8

D all of the

above

16

Jun 24-7:32 PM

14 Find the value of x.

Jun 24-7:36 PM


Jun 24-7:40 PM

16 If the m<4 = 1160 then m<9 = _____0?

n || p

17

Jun 25-3:32 PM

17 If the m<15 = 570, then the m<2 = _____0.

A 57B 123C 33D none of the above

n || p


Extending Lines to Make Transversals

131

1

410

0

Nov 27-3:45 PM

Extend the line that

would create a

transversal.

Input angle

measures

accordingly and

solve for the

missing angle.

131

1

410

0

131

1

410

0

1310

Extending Lines to Make Transversals

18


+ 12) = 180 4y + 12=x 4y + 144 = 180 4(9)+12 = x4y = 36 36 + 12 = xy = 9 x = 48

click

click

click

click

click

click

click

click


Transversals and Perpendicular Lines

Jun 25-3:40 PM

18 Find the m<1.

19

Jun 25-3:43 PM


A 12

B 54

C 42

D 18

Jun 25-3:45 PM


Jun 25-3:49 PM


20

Nov 27-4:44 PM

In the preceding section you saw that when two lines are parallel, you can conclude that certain angles created by the transversal are congruent or supplementary.

parallel.

Proving Lines are Parallel


If two lines are cut by a transversal AND the corresponding angles are congruent, then the lines are parallel.

k || mthen or or or

RememberIf two parallel lines are cut by

a transversal, then the corresponding angles are congruent.

click


Nov 27-4:05 PM


21


If two parallel lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.

If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.


click

Nov 27-4:15 PM



Theorem If two parallel lines are cut by a transversal and the alternate exterior angles are congruent,then the lines are parallel.

If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.


click

22

Nov 27-4:37 PM



Theorem If two parallel lines are cut by a transversal and the same-side interior angles are supplementary, then the lines are parallel.

m<3 + m<6 = 1800 m<4 + m<5 = 1800

If two parallel lines are cut by a transversal, then the same-side angles are supplementary.


click

Nov 27-4:59 PM


23

Jun 25-4:10 PM

22 Which statement below would make lines k and m parallel?

A m<2 = m<4

B m<5 + m<6 = 180

C m<3 = m<5

D m<1 + m<5 = 90

Jun 25-4:06 PM

23 Based on the diagram below which of the following is true?

A e || f

B f || g

C h || i

D e || g

Jun 25-4:13 PM

24 What theorem would NOT prove that lines a and b are parallel?

AIf lines a and b are cut by a transversal such that

cooresponding angles are congruent.

BIf lines a and b are cut by a transversal such that

alternate interior angles are congruent.

CIf lines a and b are cut by a transversal such that same-

side interior angles are complementary.

DIf lines a and b are cut by a transversal such that same-

side interior angles are supplementary.

24

Jul 15-2:40 PM

25 Find the value of x for which a || b.

x

Jul 15-2:47 PM

26 Find the value of x which makes a || b.

Jun 25-4:25 PM

27 Find the value of x for which m || n.

25



of Contents


Vertical ChangeRise

Run=


Slope =

Horizontal Change can be written as the difference of the x-coordinates:

26

Nov 30-2:58 PM

Undefined

Types of slopes.

Dec 3-10:41 AM

Slope of Red Line:3 - 1

0 - (-2)= 1

click

Dec 3-10:41 AM

Slope of Red Line: 2 - 10 - 3

=-1

3click

27


Red Line: 3 - 3

1 - (-2)= 0

click


Slope of Red Line:-2 - 3

=-5

0= undefined

2 - 2click

Nov 30-1:38 PM

Evaluate the slope for the given line.

: Identify two points on the

given line

(-2, -3) and (1, 3)

28

Nov 30-1:46 PM

: Evaluate the slope

using the slope formula.

: Whatever point you start with

for the y-value, you start with

the same point for the x-value.

Slope(m) =-3 - 3

-2 - 1=

-6

-3= 2

click


Evaluate the slope of the given line.

Jun 25-4:37 PM

28 How can you determine if a line has a negative slope when

looking at its graph?

AThe line goes from the bottom left to the top right of the

graph.

BThe line goes from the top left to the bottom right of the

graph.

C The line is horizontal.

D The line is vertical.

29

Jul 17-12:28 PM

29 Choose the graph(s) with a positive slope.

A

B

C

D

A. B.

C. D.

Jun 25-4:42 PM

30 Determine the slope of the given line.

Jun 25-4:44 PM

31 Determine the slope of the given line.

A 0

B 1

C -1

D undefined

30

Jun 25-4:47 PM

32 Evaluate the slope of the line containing (-2, 5) and (-5, 2).

Jun 25-4:49 PM

33 What is the slope of the line that goes through

(1, -3) and (2, -6)?

Jun 25-4:52 PM

34 What is the slope of the line that contains the points

(6, -9), and (4, -9)?

31


Writing Equations of Lines

Linear equations can be written in several different forms.

Slope-Intercept Form of a linear equation provides the slope and

-intercept of the line.

m = the slope of the line -intercept of the line

Point Slope Form of a linear equation provides the slope and a

specific point on the line.

m = the slope of the line ) = point on the line


Standard Form of a linear equation is most useful when

you want to:

-find the x & y intercepts

Ax + By = C

(A, B & C are not fractions or decimals)


Dec 3-4:25 PM

Rewrite the following equation in slope-intercept form:

The equation is in standard form and we must solve for y to put it in slope-intercept form.


Subtract 2x from both sides of the = sign

Divide both sides by 5.

32

Dec 3-4:25 PM

Identify the slope and -intercept for the following linear equation:


Hint: The equation is in standard form, if we solve it for y we

will easily be able to identify the slope and the y-intercept.click

Dec 3-4:25 PM

Write the equation of a line that has a slope of 8 and passes

through the point (-6, 7) in point-slope form.

Rewrite the same equation in slope-intercept form.


click

click

click

click

click

Jun 25-5:40 PM

35 What is the slope of the line with the given equation

5x - 2y = 15 ?

A 3

B -5/2

C 5/2

D -3

33

Jun 25-5:44 PM

36 Choose the equation of the line in point-slope form that has a

slope of 1/4 and contains the point (-2, 7).

A y + 7 = 1/4 (x + 2)

B y = 1/4 x + 7.5

C y - 7 = 1/4 (x - 2)

D y - 7 = 1/4 (x + 2)

Jul 2-3:46 PM

37 Write the equation of the line in slope-intercept form

that contains the following points: (4,3) and (-8,6)

A y = -1/4 x - 1

B y = -1/4 x + 2C y=-1/4 x +4D y = -1/4 x + 7

Jun 25-5:48 PM

38 Determine the equation of the line in slope-intercept form.

A y = 3/4 x + 3

B y = 3/4 x - 4

C 3x - 4y = -12

D y = 4/3 x + 3

34

Lab 2


To investigate slopes of parallel lines go to the lab

titled,"Slopes of Parallel Lines".

Click here to go to Slopes of

Jul 3-1:13 PM


Write a conjecture about slopes of parallel lines.


Two lines have the same slope if and only if they are parallel.


certain they are parallel, we

must evaluate the slope of

each line.

Red Line: (1, 2) and (5, 4)

Blue Line: (0, -2) and (4, 0)

The slopes of the two lines are the same.

Therefore they are parallel.

m = 5 - 1

= 1/24 - 2

m = 0 - (-2)

4 - 0= 1/2

click

click

35




1. Determine whether LM and NO are parallel given the following information:

L: (3,-5) M: (-6,1) N: (4,-5) O: (7,-7)

-6 - 3

1 - (-5)= -2/3m=

: Evaluate the slope for each line

-7 - (-5)

7 - 4= -2/3m=

: Compare the slopes of each line to determine if they

are parallel.


click click


: Identify the information given in the problem.

: Identify what information you still need to create the

equation and choose the method to obtain it.

The slope: Use the equation of the parallel line to determine the

Therefore m = 3/2


click

clickclick

36

Dec 3-4:18 PM

: Create the equation.

Point-Slope Form

Slope-Intercept Form

The correct solution to the original problem is either form of the

equation. Point-Slope Form and Slope-Intercept Form are two

ways to write the same linear equations.


click

click

click

Jun 25-5:56 PM

39 What is the equation of the line passing through (6, -2) and

parallel to the line whose equation is y = 2x - 3?

A y = 2x + 2

B y = -2x + 10

C y = 1/2 x - 5

D y = 2x - 14

Jun 25-5:58 PM

40 Which is the equation of a line parallel to the line represented

by: y = -x - 22 ?

A x - y = 22

B y - x = 22

C y + x = -17

D 2y + x = -22

37

Jun 25-6:02 PM

41 Two lines are represented by the equation:

-3y=12x-14 and y=kx+14

For which value of k will the lines be parallel?

A 12

B -14

C 3

D -4

Jun 25-6:06 PM

42 Which equation represents a line parallel to the line whose

equation is:

3y + 4x = 21

A 12y + 16x = 12

B 3y - 4x = 22

C 3y = 4x + 21

D 4y + 3x = 21


43

38


44 What is an equation of the line that passes through the point (5,-2) and is parallel to the

Perpendicular Lines & Angles

Perpendicular Lines

of Contents

Dec 3-5:01 PM

Two lines are perpendicular if and only if they form 4

right angles when they intersect.

Perpendicular Lines

Perpendicular lines are also coplanar.

39


In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.

click

click

Perpendicular Lines


45


46

40


47


48


49

AB || DF

41



of Contents

Dec 4-1:09 PM

Dec 4-1:15 PM

Slopes of Perpendicular LinesNegative Reciprocals combine opposites and reciprocals. Finish filling out the table below.

Original

NumberOpposite Reciprocal

Negative

Reciprocal

A. -1/4

B. C.

D. E. F. 2/3

H. I.

42


50


51

Lab 3


To investigate slopes of perpendicular lines go to the lab

titled,"Slopes of Perpendicular Lines".

Click here to go to Slopes ofPerpendicular Lines

43

Jul 3-2:27 PM


Write a conjecture on slopes of perpendicular lines.


The red line rises, so it has a positive slope.

Slope of Blue Line =

Slope of Red Line =

Observations

-2/3

3/2

click

click

Perpendicular lines have negative reciprocal slopes.

What do you notice about the lines?


redblue


their

Slope of Blue Line = 1 Slope of Red Line = -1


click click

redblue

44

Dec 4-1:54 PM

Slope of Blue Line

Slope of Red Line

= 4/5

= -2/3


click

click

red

blue


Any horizontal line and vertical are always perpendicular

because they form 4 right angles at their point of intersection.



52 Are these two lines perpendicular?

45


53


54


55

46

Dec 4-2:09 PM


Facts about slope that can assist with writing linear equations:

-Parallel Lines have the SAME slope

-Perpendicular Lines have NEGATIVE RECIPROCAL slopes

one another

click

click

click

click

click


: Identify the slope according to the given equation.

Given equation: m = 1/2 Perpendicular Line: m = -2

Point-Slope Form

Slope-Intercept Form

y - 5 = -2 (x + 2)

y - 5 = -2x - 4

y = -2x + 1


click click

click

click

click

Dec 4-2:22 PM


47


56

the line whose equation is 4


57 What is an equation of the line that contains the

equation is


58

What would be the best statement to describe these two lines?

=15 5(

48

Proofs

Proofs Involving Parallel

Perpendicular Lines

of Contents

Jul 3-3:56 PM

Proofs in Geometry

A proof is a logical list of steps that is used to reach a conclusion. Each step is supported by a theorem, postulate, definition, or property.

There are three types of proof.

1. 2-column proof2. flowchart proof3. paragraph proof

Jul 3-4:46 PM

Proofs in Geometry

Example of a 2 column proof Proof of Same-Side Interior Angles Theorem.

Given: a || bProve: <2 and <3 are supplementary

a

b

12

3

NOTE: If we are going to prove Same-Side Interior Angles Theorem we cannot use it as a reason in our proof.

Statement Reason

3. m<1+m<2=180

0

4. <1 ≅<3

1. a || b

6. m<3+m<2=180

1. Given

2. A linear pair of angles is supplementary. (Linear Pair Postulate)

4. Alternate Interior Angles Theorem

6. Substitution

7. <3 and <2 are supplementary

7. Definition of supplementary

2. <1 and <2 are supplementary

0 3. Def of supplementary

5. m<1=m<3 5. Def. of Congruent Angles

49

Jul 5-11:02 AM

Proofs in Geometry

Example of a flowchart proof Proof of Same-Side Interior Angles Theorem.


a

b

12

3


a || b

m<1+m<2=180

<1≅<3

m<3+m<2=180<3 and <2 are supplementary

0

0

givenLinear PairPostulate Substitution

Def. of supplementary

<1 and <2 aresupplementary Def. of

Supplementary

Alt. Int. <'sTheorem

m<1=m<3

Def. of congruentangles

Jul 5-11:02 AM

Proofs in Geometry

Example of a paragraph proof Proof of Same-Side Interior Angles Theorem.


a

b

12

3


Because a || b we know that <1≅<3 by alternate interior angles

theorem, which also makes m<1 = m<3 by def. of congruent angles. We also know that <1 and <2 are supplementary by linear pair postulate, which implies that m<1 + m<2 = 180 by definition of supplementary. By substitution m<3 + m<2 = 180. Therefore, <3 and <2 are supplementary by definition of supplementary.

0

0

Jul 5-11:31 AM

Proofs in Geometry

The justifications in a proof are made up of definitions, theorems, postulates, and properties.

Theorems and Postulates

Vertical Angles Theorem

Linear Pair Postulate

Parallel and Perpendicular Postulate

Corresponding Angles Theorem

Alternate Interior Angles Theorem

Same-Side Angles Theorem

Alternate Exterior Angles Theorem

Converse of Corresponding Angles Thm

Converse of Alternate Interior Angles Thm

Converse of Same-Side Angles Thm.

Converse of Exterior Angles Thm.

Segment Addition Postulate

Angle Addition Postulate

Congruent supplements / compliments thm

50

Proofs

Proofs in Geometry

Dec 5-9:45 AM

Substitution Property , then either a or b may be

substituted for the other in any equation/inequality

Reflexive Property

Symmetric Property

Distributive Property

Proofs in Geometry

Proofs

Reflexive Property:

Symmetric Property: If , then

If , then

Transitive Property: If and , then

If and , then

Proofs in Geometry

51

Dec 5-10:07 AM

Reason

3)

4) Transitive Property

Proofs in Geometry

Given: k || mProve: Alternate Interior Angles Theorem

NOTE: If we are going to prove Alternate Interior Angles Theorem we cannot use it as a reason in our proof.

Complete the proof.

Proofs

0

Reasons

0

0

Proofs in GeometryUnscramble the list of reasons in the following proof.

a) Same-Side Angles Thm.

b) Substitution Property

c) Given

d) Vertical Angles Thm.

e) Def. of Congruent Angles

Jul 5-12:13 PM

Proofs in Geometry

Supply the missing reasons in the following proof.

Prove vertical angles theorem.

12

34

NOTE: If we are going to prove Vertical Angles Theorem we cannot use it as a reason in our proof.

m<1+m<2 = 180

m<2+m<3=180

m<1= m<3

0

0

1. _____________

2. ____________

7. ____________

<1 and <2 forma linear pair

<2 and <3 forma linear pair

<1 and <2 aresupplementary

<2 and <3 aresupplementary

3. ____________

4. ____________

5. ____________

6. ____________

52

Jul 5-12:30 PM

Proofs in Geometry

Fill in the blanks in the following paragraph proof.

Given: m<2 = 125, m<4 = 55

Prove: k || m

0 0

It is given that m<2 = 125 and m<4 = 55 and we know that 125 + 55 = 180 which implies that m<2 + m<4 = 180 by a)____. If m<2 + m<4 = 180 then <2 and <4 are supplementary by definition of b._____. If <2 and <4 are supplementary them k || m by the converse of c._____.

0 0 0

00 0

Jul 9-11:08 AM

Proofs in Geometry

Given: <AProve: ABCD is a parallelogram

Statements Reasons

1. <A≅<C; <B≅<D

3. m<A+m<B+m<C+m<D= 360

4. m<A+m<B+m<A+m<B=

3600

5. 2(m<A+m<B) = 3600

6. m<A+m<B = 1800

7. <A and <B are supplementary

8. <C and <D are supplementary

9. BC || AD; AB || CD

10. Quad ABCD is a parallelogram

i. given

h. The sum of the interior angles of a

quad is 3600

g. Substitution property

d. Distributive Property

b. Division property of equality

f. def. of supplementary angles

a. substitution property

e. converse of the same side angles theorem.

c. Definition of parallelogram

2. m<A=m<C; m<B=m<Dj. def. of congruence

0

Unscramble the reasons in the following proof.

Jul 9-11:08 AM

Proofs in Geometry

Given: BD || AC

Prove: m<2+m<4+m<5 = 1800

Supply the missing reasons in the following proof.

3. m<1=m<4; m<5=m<3

4. m<1+m<2+m<3=18000

5. m<4+m<2+m<5=1800

1. BD || AC

__ __

Statements Reasons

1.

2.

3.

4.

1. BD || AC

2. <1=<4; <5=<3~ ~

5.

53

Jul 5-3:20 PM

59 If a || b, how can we prove m<1=m<4?

A Corresponding angles postulate

B Converse of corresponding angles postulateC Alternate Interior angles theoremD Converse of alternate interior angles theorem

ab

c

1 4

3

2

Jul 5-3:27 PM

60 If m<1=m<3, how can we prove a || b?

A Corresponding angles postulate

B Converse of corresponding angles postulateC Alternate Interior angles theoremD Converse of alternate interior angles theorem

ab

c

1 4

3

2

Dec 5-10:51 AM

61 What is the justification for the missing component

in the provided proof?

A Alternate Interior

Angles Theorem

B Same-Side Interior

Angles Theorem C Corresponding

Angles Theorem

D Given

Given:Prove:

, AB ||DE

Justification

1) 1)

2)

3)

5)

2)

3)

5)

6)of BDC

6)


Corresponding Angles Postulate

Transitive

Def. of bisector

, AB ||DE

4) 4) Transitive

54

Dec 5-10:36 AM

62 What is the justification for the missing component

in the provided proof?

Angles

Converse Theorem Angles

Converse Theorem

Angles

Converse Theorem Definition of

Supplementary Angles

Justification

vertical angles thm

Transitive

Dec 5-10:47 AM

63 What is the missing statement in the provided proof?

A B

C D

Given:Prove: k || m

Justification

1) Given

k || m

Def. perpendicular Lines & Def. of

right angle

5)

6)

5)

6)

Def. perpendicular Lines & Def. of

right angleSubstitution

Corresponding Angles Converse

Theorem

Def. of Congruent Angles

Jul 5-1:59 PM

64 Given m<1=m<2, m<3=m<4, what can we prove?

A a || b

B c || dC line a is perpendicular to line c D line b is perpendicular to line d

a

b

d

1

23

4 5 c

55

Jul 5-2:05 PM

65 Given c || d, what can we prove?

A m<1 = m<2

B m<4 = m<5C m<2 = m<3D m<2 + m<5 = 180

a

b

d

1

23

4 5 c



of Contents

Jul 9-4:37 PM

Parallel Line Construction

Constructing geometric figures means you are constructing lines, angles, and figures with basic tools accurately.

We use a compass, and straightedge for constructions, but we also use some paper folding techniques.

Construction by: MathIsFun

56

Jul 9-4:45 PM


Have students do constructions with the following slides.


When asked to construct a line through a point parallel to a given line, there are

Method 1

Given


Dec 5-11:11 AM

Construction Continued

0°

34

0°24

57


: Mark the arc intersection

point E and use a ruler to join C

therefore

0°24

Video

Video Demonstrating Constructing Parallel Lines with Corresponding Angles using Dynamic Geometric Software

Click here to see video


Given

58

Dec 5-11:43 AM

0°

24

0°24


: Mark the arc intersection point E

and use a ruler to join C and E.

therefore

0°24

Video

Video Demonstrating Constructing Parallel Lines with Alternate Interior Angles

using Dynamic Geometric Software


59


Method 3

Given

Dec 5-11:51 AM

0°24

0°24

Dec 5-11:51 AM

: Mark the arc

intersection point E and use a

ruler to join C and E.

therefore

0°24

60

Video

Video Demonstrating Constructing Parallel Lines with Alternate Exterior

Angles using Dynamic Geometric Software


Jun 26-12:52 PM

Parallel Line Construction Using Patty Paper

Step 1: Draw a line on your patty paper. Label the line g. Draw

a point not on line g and label the point B.

Step 2: Fold your patty paper so that the two parts of line g lie

exactly on top of each other and point B is in the crease.

Jun 26-1:09 PM

Step 3: Open the patty paper and draw a line on the crease.

Label this line h.

Step 4: Through point B, make another fold that is perpendicular

to line h.

Parallel Line Construction Using Patty Paper

61

Jun 26-3:21 PM

Step 5: Open the patty paper and draw a line on the crease.

Label this line i.

Because lines i and g are perpendicular to line h they are

parallel to each other. Therefore line i || line g.

Video

Video Demonstrating Constructing a Parallel Line using Menu Options of Dynamic Geometric Software

Click here to see video 2

Click here to see video 1

Dec 5-12:02 PM

66 The lines in the diagram below are parallel because





62

Dec 5-12:02 PM






Dec 5-12:09 PM





Corresponding Angles Postultate



of Contents

63

Jul 9-4:53 PM

Perpendicular Line Construction

Have students do constructions with the following slides.

Construction by: MathIsFun


Construct a line through a point perpendicular to a given line.

Constructing a Perpendicular Line

0°

118

Dec 4-3:30 PM

Construction Continued

0°

118

64


Name the point of intersection of the two arcs as F.


Example

Dec 4-3:56 PM

Example Continued

: From points A and B, draw 2 intersecting arcs below line using the same compass width.

65

Dec 4-3:51 PM

Video

Video Demonstrating Constructing a Perpendicular Line with a Compass and Straightedge using Dynamic Geometric

Software


Jun 26-3:30 PM

Constructing a Perpendicular Line Using

Patty Paper

Step 1: Draw a line on your patty paper. Label the line g. Draw a

point not on line g and label the point B.

Step 2: Fold your patty paper so that the two parts of line g lie

exactly on top of each other and point B is in the crease.

66

Jun 26-3:34 PM

Step 3: Open the patty paper and draw a line on the crease. Label

this line h.

Line h is perpendicular to line line g.

Video

Video Demonstrating Constructing a Perpendicular Line using Menu Options of

Dynamic Geometric Software


Jul 9-6:00 PM

Try This:

1. Construct a 60 angle

2. Construct a regular hexagon in a circle.

Constructions

Click here to view theanimated construction ofa 60 angle.

0

Construction by: Mathisfun

Click here to view theanimated construction ofa hexagon inscribed in a circle.

Construction by: MathOpenReference

67

Video

Videos Demonstrating Constructing a 60 Degree Angles and a Hexagon using

Dynamic Geometric Software

Click here to see60 degree angle video

Click here to seehexagon video


69 Which diagram represents the construction of the

perpendicular bisector of MN?

geometry parallel and perpendicular · pdf fileparallel and perpendicular lines geometry ......

Documents