Math II – Parallel Lines Task NAME PERIOD...

Math II – Parallel Lines Task NAME_______________________PERIOD____________SCORE_________

1. Draw two parallel lines with at least 2 inches between them and at least 4 inches long.

2. Construct a diagonal line that passes through both parallel lines. This diagonal line is called a TRANSVERSAL.

3. Number each of the angles formed using #1 - #8.

4. Measure each angle #1 - #8 with a protractor and list each measure below.

<1 = <2 = <3 = <4 = <5 = <6 = <7 = <8 =

5. List EVERYTHING you notice about specific angles. List below using the angle’s specific number.

Unit 3 Notes

Math II Unit 3.1 Parallel Lines,Transversals and Triangles

1. Pass out the parallel line and transversal TASK.

2. Discuss findings about parallel lines and transversals.

1 23 4

5 6

7 8

m

n

p

3. Vocabulary:

a) Alternate interior angles -

b) Alternate exterior angles -

c) Vertical angles -

d) Consecutive interior angles -

e) Corresponding angels -

f) Linear pair -

1 2

3 4

5 678

4. Find the measure of each missing angle if

213 4

5 67 8

5. Find each angle.

b

cd

e

21150

34 5

6 78

9750

1112

131415

16

1718

19 20

6.TASK: Construct any triangle and measure each angle. Construct a different triangle and measure each angle.

a)The sum of the measures of the interior angles of a triangle =

7. Find each missing angle.

A

B C450880

x

a) b)

A

BC1100x

y

650

Unit 3 Notes

8. Find each missing angle. Side/Angle Relationships:

The largest angle is always opposite the longest side. The smallest angle is always opposite the shortest side.

List the sides in order, smallest to largest

List the angles in order, smallest to largest

a

b

c99

46

35

Z

Y

X

17.3 in

15 in

9.7 in

9. Find 10. Isosceles Triangles -

If , find

If , find

In the figure,

and

BL

R P

Unit 3 Notes

Isosceles Triangle:

At least 2 sides (called the legs) of the triangles are congruent.

Base Angles

Base angle theorem: The base angles of an isosceles triangle are congruent.

Definitions:

Quadrilateral - A polygon with four vertices and four edges.

Parallelogram - A quadrilateral with both pairs of opposite sides parallel.

Woot Star Trek

Which of the following are quadrilaterals? To show opposite sides of a parallelogram P are congruent, which triangles would youshow are congruent?

P A

G R

M

Unit 3 Notes

Use PGR and RAP in the parallelogram from Question 3 to prove that oppositesides of a parallelogram are congruent . Prove the statement and .

Given: Parallelogram PARG with diagonals PR and AG intersecting at point MProve: and

P A

G R

M

Now we can say for certain that:

In a parallelogram opposite sides are congruent

Our next parallelogram theorem tell us that:

Opposite angles of a parallelogram are congruent.

P A

G R

P A

G R

M

The next parallelogram theorem tells us that:Diagonals bisect each other, which means that the opposite sides are congruent.

Unit 3 Notes

This theorem takes 2 seconds to prove

Given: PARG is a parallelogram

Prove:

P A

G R

Recap: The 5 things we know about Parallelograms

In a parallelogram...

Opposite sides

xo (180-x)o

Opposite sidesare congruent are parallel Opposite angles

are congruent

Diagonals bisecteach other

Consecutive anglesare supplementary

Find the value of each variable in the parallelogram.

3y-7

3x+18

5x

2y+4

Find the value of each variable in the parallelogram.

(3x-18)o4yo

(2x+12)o 3zo

Unit 3 Notes

P A

G R

M

Use what you know about parallelograms to find the length of all the sides.

3x + 7

12x - 11

3y + 9y - 11

8z + 10

34

12

4t + 8

There are other converse theorems to prove that a quadrilateral is a parallelogram. We don't have time to prove them all. You will do one of them in your homework.

If opposite sides are congruent...

If opposite angles are congruent...

If the diagonalsbisect each other...

If opposite sides are congruent and parallel...

...then the quadrilateral is a parallelogram.

Are you given enough information to determine whether the quadrilateral is a parallelogram? (Remember what it is labeled is more important then what it looks like.)

What do we know about Parallelograms?

Unit 3 Notes

Also remember from last time:

If opposite sides are congruent...

If opposite angles are congruent...

If the diagonalsbisect each other...

If opposite sides are congruent and parallel...

...then the quadrilateral is a parallelogram.

These will make things easy today

Definition:

Rhombus - a quadrilateral with all sides congruent.

Prove that rhombus RHOM is a parallelogram. (if you remember last time this should take one statement)

R H

OM

Since a rhombus is a parallelogram, what properties hold true for all rhombi?

Unit 3 Notes

The next theorem for Rhombi is:The diagonals of a Rhombi are Perpendicular.

What does this mean?

R H

OM

B

We also can prove that the diagonals of a rhombus bisect the vertex angles. What does that mean?

Use what we know about Rhombi to find the following: A B

CD

E

13y + 8

3x + 12

4x - 2

10y + 62

11

Definition:

Rectangle - A quadrilateral with all angles congruent.

Unit 3 Notes

Prove that rectangle RECT is a parallelogram.

R E

C T R E

CT

A

Rectangle)Theorem:

Diagonals)bisect)each)other.

)All)sec6ons)are)congruent.)

Definition:

Square - A quadrilateral with all four sides and all four angles congruent.

Venn Diagram: Put all the quadrilaterals we learned so far in the appropriate place in the venn diagram

Quadrilaterals Parallelograms SquaresRectangles Rhombi

Unit 3 Notes

Now that we know that a square is a parallelogram, a rhombus, and a rectangle, what properties does a square have?

ABCD is a square solve for x

5xo

19

A B

CD

FGHI is a Rectangle. Solve for x.

F

G H

I

FH = 8x-13GI = 7x+11

Math II Parallel Lines CW NAME__________________________PERIOD___________SCORE___________ Use$the$given$information$to$determine$the$measures$of$each$of$the$numbered$angles.$!

1. p q !and! 1 54m∠ = o!!2345678

mmmmmmm

∠ =∠ =∠ =∠ =∠ =∠ =∠ =

!!

!2. Suppose!that!two!parallel!lines!are!intersected!by!a!transversal!and!all!

corresponding!angles!are!supplementary.!!How!is!this!possible?!Sketch!and!label!a!figure!to!support!your!answer.!

!

3. Angles!2!and!8! ! ! ! ! 4.!Angles!1!and!4!

5. Angles 6 and 7 6. Angles 4 and 5

Find the measure of each missing angle.

7. 8.

Chapter 4 16 Glencoe Geometry

PracticeAngles of Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-2

Copyright ©

Glencoe/M

cGraw

-Hill, a division of T

he McG

raw-H

ill Com

panies, Inc.

Find the missing angle measures.

1. 2.

Find the measure of each angle.

3. m!1

4. m!2

5. m!3


6. m!1

7. m!4

8. m!3

9. m!2

10. m!5

11. m!6

Find the measure of each angle if !BAD and !BDC are right angles and m!ABC ! 84.

12. m!1

13. m!2

14. CONSTRUCTION The diagram shows an example of the Pratt Truss used in bridgeconstruction. Use the diagram to find m!1.

145"1

64"1

2A

BC

D

118"36"

68"

70"

65"

82"

1

2

3 4

5

6

58"

39"

35"

12

3

40" 55"

72"

?

05-56 Geo-04-873961 4/3/06 4:54 PM Page 16


PracticeAngles of Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-2

Copyright ©

Glencoe/M

cGraw

-Hill, a division of The M

cGraw

-Hill C

ompanies, Inc.

Find the missing angle measures.

1. 2.


3. m!1

4. m!2

5. m!3


6. m!1

7. m!4

8. m!3

9. m!2

10. m!5

11. m!6

Find the measure of each angle if !BAD and !BDC are right angles and m!ABC ! 84.

12. m!1

13. m!2

14. CONSTRUCTION The diagram shows an example of the Pratt Truss used in bridgeconstruction. Use the diagram to find m!1.

145"1

64"1

2A

BC

D

118"36"

68"

70"

65"

82"

1

2

3 4

5

6

58"

39"

35"

12

3

40" 55"

72"

?

05-56 Geo-04-873961 4/3/06 4:54 PM Page 16

Math II Parallelograms CW NAME_______________________________PERIOD__________SCORE_______

Determine whether each quadrilateral is a parallelogram. Justify your answer.

10. 11.

12. 13. 14.


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PracticeTests for Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

6-3


1. 2.

3. 4.

COORDINATE GEOMETRY Determine whether a figure with the given vertices is a parallelogram. Use the method indicated.

5. P(!5, 1), S(!2, 2), F(!1, !3), T(2, !2); Slope Formula

6. R(!2, 5), O(1, 3), M(!3, !4), Y(!6, !2); Distance and Slope Formula

ALGEBRA Find x and y so that each quadrilateral is a parallelogram.

7. 8.

9. 10.

11. TILE DESIGN The pattern shown in the figure is to consist of congruent parallelograms. How can the designer be certain that the shapes areparallelograms?

!4y ! 2

x " 12

!2x " 6

y " 23!4x " 6

!6x

7y " 312y ! 7

3y ! 5!3x " 4

!4x ! 22y " 8(5x " 29)#

(7x ! 11)#(3y " 15)#

(5y ! 9)#

118# 62#

118#62#

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.PracticeTests for Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

6-3


1. 2.

3. 4.





7. 8.

9. 10.


!4y ! 2

x " 12

!2x " 6

y " 23!4x " 6

!6x

7y " 312y ! 7

3y ! 5!3x " 4

!4x ! 22y " 8(5x " 29)#

(7x ! 11)#(3y " 15)#

(5y ! 9)#

118# 62#

118#62#

Less

on 6

-3

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NAME ______________________________________________ DATE ____________ PERIOD _____

6-3


1. 2.

3. 4.





7. 8.

9. 10.


!4y ! 2

x " 12

!2x " 6

y " 23!4x " 6

!6x

7y " 312y ! 7

3y ! 5!3x " 4

!4x ! 22y " 8(5x " 29)#

(7x ! 11)#(3y " 15)#

(5y ! 9)#

118# 62#

118#62#

Less

on 6

-3

05-56 Geo-06-873963 4/3/06 2:06 PM Page 23


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NAME ______________________________________________ DATE ____________ PERIOD _____

6-3


1. 2.

3. 4.





7. 8.

9. 10.


!4y ! 2

x " 12

!2x " 6

y " 23!4x " 6

!6x

7y " 312y ! 7

3y ! 5!3x " 4

!4x ! 22y " 8(5x " 29)#

(7x ! 11)#(3y " 15)#

(5y ! 9)#

118# 62#

118#62#

Less

on 6

-3

05-56 Geo-06-873963 4/3/06 2:06 PM Page 23


Cop

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NAME ______________________________________________ DATE ____________ PERIOD _____

6-3


1. 2.

3. 4.





7. 8.

9. 10.


!4y ! 2

x " 12

!2x " 6

y " 23!4x " 6

!6x

7y " 312y ! 7

3y ! 5!3x " 4

!4x ! 22y " 8(5x " 29)#

(7x ! 11)#(3y " 15)#

(5y ! 9)#

118# 62#

118#62#

Less

on 6

-3

05-56 Geo-06-873963 4/3/06 2:06 PM Page 23


Cop

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NAME ______________________________________________ DATE ____________ PERIOD _____

6-3


1. 2.

3. 4.





7. 8.

9. 10.


!4y ! 2

x " 12

!2x " 6

y " 23!4x " 6

!6x

7y " 312y ! 7

3y ! 5!3x " 4

!4x ! 22y " 8(5x " 29)#

(7x ! 11)#(3y " 15)#

(5y ! 9)#

118# 62#

118#62#

Less

on 6

-3

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Math II 3-4 Special Parallelograms CW NAME____________________________PERIOD__________SCORE________

Each figure is a parallelogram. Identify the special type and explain your reasoning.

1 2

3

4

5 6

7 8

9 10

11

Math II – Parallel Lines Task NAME PERIOD...

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