Butterflies, Pinwheels, and Wallpaper · Butterflies, Pinwheels and Wallpaper Investigation #3 Transforming Coordinates Investigation #4 Dilations and Similar Figures Problem 3.1:

Butterflies,

Pinwheels, and

Wallpaper

Investigation #3: Transforming Coordinates Investigation #4: Dilations and Similar Figures

Name________________________________________________

M8 – BPW – Investigation 3 and 4

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Butterflies, Pinwheels and Wallpaper

Investigation #3

Transforming Coordinates

Investigation #4

Dilations and Similar Figures

Problem 3.1: How can you describe how points “move” under a reflection with coordinate

rules in the form (x, y) → ( , ) when the reflection line is: (1) the x-axis? (2) the y-axis?

(3) the line y = x?

Problem 3.2: What kind of coordinate rule (x, y) → ( , ) tells how to “move” any point to its image under a translation?

Problem 3.3: What are the coordinate rules that describe “motion” of points on a grid under

turns of 90° and 180°?

Problem 3.4: How are lines and their images under translations and half-turns related to

each other?

Problem 3.5: When two parallel lines are cut by a transversal, what can be said about the

angles formed? What is always true about the angle measures in a triangle? How do you

know that your answers are correct?

Problem 4.1: What coordinate rules model dilations? How do dilations change or preserve

characteristics of the original figure?

Problem 4.2: How can you use transformations to check whether two figures are similar or

not?


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Reflecting over the y-axis:


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Are the two figures congruent? Explain.


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Reflecting over the x-axis:


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Practice:

1. Quadrilateral ABCD is being reflected over the

x-axis.

Draw and label the image quadrilateral A’B’C’D’.

We can use an arrow to describe this reflection.

ABCD A’B’C’D’

What are the coordinates of ABCD ?

A _______ A’ _________

B _______ B’ _________

C _______ C’__________

D________ D’__________

2. Describe a reflection that would move shape 1 to match shape 2.


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Key Concepts:

A transformation is a change in the ___________________, ____________________,

or _____________________ of a figure.

A reflection is a transformation which __________________ the figure over a

____________________.

This line is called the .

The original image of a transformation is called the __________________________.

The resulting figure after a transformation is called the _________________ of the

original figure, and is labeled with _____________.

Example:

If a reflection occurs the image and pre-image are _________________________

because the ____________________ and _______________________ are persevered.


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H

O

T

1 2 3 4 5 6 7–1–2–3–4–5–6–7 x

1

2

3

4

5

6

7

–1

–2

–3

–4

–5

–6

–7

y

Homework:

1. a. Draw and label the image of the triangle HOT after a reflection over the x-

axis.

b. Write the coordinates of H’O’T’.

c. Is the image congruent to the pre-

image? How do you know?

2 a. Draw and label the image of t quadrilateral WXYZ after a reflection over the

dotted line.

Label the image W’ X’ Y’ Z’.

b. Are the two figures congruent?

c. What is the equation of the dotted line?

W

X

Y

Z

1 2 3 4 5 6 7 8–1–2–3–4–5–6–7–8 x

1

2

3

4

5

6

7

8

–1

–2

–3

–4

–5

–6

–7

–8

y


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3. The table below shows the coordinates of triangle PQR. Reflect PQR over the y-

axis. Complete the table for the coordinates of the image.

Triangle PQR

Triangle P’Q’R’

P (-3, 2) P’

Q (3, 6) Q’

R (-7, 7) R’

Explain how the coordinates of R changed to

the coordinates of R’.

4. a) Draw ΔJKL which has coordinates J (-2,0), K (-3,4), and L (5,-1).

b) Draw the image ΔJ’K’L’ after a reflection of ΔJKL over the x-axis.

c) List the coordinates of J’K’L’.

J (-2,0) J’ _________

K (-3, 4) K’ _________

L (5, -1) L’__________

d) How did the coordinates of K change

to the coordinates of K’ ?

e) Are the two figures congruent? Explain.


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Ex:) Given Δ𝐴𝐵𝐶 and image Δ𝐴′𝐵′𝐶′, list the coordinates of image Δ𝐴′𝐵′𝐶′.

A ______ A’ _________

B _______ B’ _________

C _______ C’__________

Write the rule that translates Δ𝐴𝐵𝐶 to its image

Rule: (𝑥 , 𝑦) ( ___________, ____________)


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Part A: Make a table of the coordinates of key

points for Mug 1 and his images under the translations. Look for patterns:

4. Are all the mugs congruent? Explain.

E. Suppose a translation moves a figure a units horizontally and b units vertically

on a coordinate grid. What rule describes the coordinates of each image point?


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Key Concepts:

A translation is a transformation which ___________________ each point of a figure

the same ________________________ and in the same ____________________________.

If a translation occurs the image and pre-image are _________________________


Practice:

1. Write a general rule which describes the translation shown below. ΔLMN is the

original triangle.

In words: ____________________________________

Arrow Notation: (x, y) ( , )

2. a) Graph points T(0,3), U(2, 4) and V(5, -1) and

connect the points to make a triangle.

b) Translate ΔTUV using the rule (x, y) (x - 3, y – 1).

c) In words, describe what the rule is asking you to do.

d) Draw the image ΔT’U’V’.

e) Identify the coordinates of ΔT’U’V’. T’ U’ V’

f) Using the image of ΔT’U’V’ perform an additional translation using the rule:

(x, y) (x + 3, y - 3). Draw and Label ΔT’’U’’V’’

g) Is this new image (ΔT’’U’’V’’) congruent or similar to the original figure (ΔTUV) ?

Explain.


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Practice

In questions a and b below, use arrow notation to write a rule (in words or arrow

notation) that describes the translation shown on the graph.

1) 2)

3) Multiple Choice:

Write a description of the rule. (x, y) (x – 7, y + 4)

1. translation 7 units to the right and 4 units up.

2. translation 7 units to the left and 4 units down.

3. translation 7 units to the right and 4 units down.

4. translation 7 units to the left and 4 units up.

4) Refer to the grid below.

a. Describe how you could move shape 1 to exactly

match shape 2 by using one translation and one

reflection.

b. Are there other sequences of transformations that

would move shape 1 to exactly match shape 2? If so, describe the steps you would

perform.


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H

O

T

1 2 3 4 5 6 7–1–2–3–4–5–6–7 x

1

2

3

4

5

6

7

–1

–2

–3

–4

–5

–6

–7

y

W

X

Y

Z

1 2 3 4 5 6 7 8–1–2–3–4–5–6–7–8 x

1

2

3

4

5

6

7

8

–1

–2

–3

–4

–5

–6

–7

–8

y

Homework

1. On the graph below, draw and label the image of triangle HOT after a

translation six units left and one unit down. Is the image similar or

congruent? How do you know?

2. On the graph below, draw and label the image of quadrilateral WXYZ under

the rule 4,2),( yxyx .


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A

B

C

D

E

2 4 6 8–2–4–6–8 x

2

4

6

8

–2

–4

–6

–8

yUse the grid below to answer

questions 3 through 5.

3. Find the rule to describe

the translation from point A to

point B.


the translation from point C to

point D.


the translation from point E to

point A.

6. Quadrilateral ABCD is plotted on the grid below.

Part A: On the graph, draw the

translation of polygon ABCD

eight units to the left and seven

units down. Label the image

A’B’C’D’.

A ______ A’ _________

B _______ B’ _________

C _______ C’__________

D _______ D’__________

Part B: Explain how you

determined the location of A’.


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S

P Q

RO x

y

7. Quadrilateral PQRS is plotted on the grid.

On the graph, draw the translation of polygon

PQRS three units to the left and four units down.

Label the image P’Q’R’S’.

Now create polygon P”Q”R”S” by translating

polygon P’Q’R’S’ using the rule

1,2, yxyx . What will be the coordinates of

point Q”?

Answer ______________

Write a single translation rule from polygon PQRS to

polygon P”Q”R”S”.

Triangle ABC is translated to create triangle DFG, as shown below:

In these triangles, side AB is congruent to side DF, and side BC is congruent to side

FG. Determine the values of x and y.


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D

H

FD'

H'

F'

O x

y

D

H

F

D'

H'

F'

O x

y

3. Rotate Triangle ABC, 270o counterclockwise.

Label the triangle A′′′ B′′′ C′′′.

4. Graph ABCD with vertices A(0, 4), B(2, 2), C(4, 2), and D(4, 4).

What are the coordinates of the image of quadrilateral ABCD under a 90°

clockwise rotation about the origin?

A (0, 4) A’ _________

B (2, 2) B’ _________

C (4, 2) C’__________

D (4, 4) D’__________

5. Determine the transformation that produced the following images:

a) b)


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Key Concept:

A turning of a geometric figure about a point is a _________________________. The

_________________________________________ is the point about which the figure is

turned. The number of degrees the figure turns is the _________________ of rotation.

If a rotation occurs the image and pre-image are _________________________


Summary:

A _____________ ________________ is when a reflection, translation or rotation

occurs.

If two figures are Congruent then the ________________________ and

__________________________ are preserved.

If a _____________________, ____________________________, ______________________

occur, then the two figures are congruent.


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1. a) Graph Triangle RST with vertices

R 3,2 , S ,4,5 and T .8,4

b) Rotate Triangle TSR 90° counterclockwise on

the graph to the right. Label the new coordinates

of the vertices of Triangle TSR . Graph the image.

R (-2,3) R’ _________

S (5,4) S’ _________

T (4,8) T’__________

c) Are RST and TSR congruent to each other,

similar to each other, or neither? Explain.

2. a.) Graph Triangle ABC, A(3, –2), B(–5, 4),

C( 3, 2).

b) Graph and label the image of Triangle

ABC after a 180° counterclockwise rotation

about the origin.

A (3, –2) A’ _________

B (–5, 4) B’ _________

C (3, 2) C’__________


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3. Quadrilateral ABCD is plotted on the grid below.

Part A

On the graph, draw the image of quadrilateral ABCD after a counterclockwise

rotation of 180o about the origin. Label the image DCBA . List the coordinates

below.

A A’ _________

B B’ _________

C C’ _________

D D’ _________

Part B

Explain how the coordinates of A changed to the coordinates of A .


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4. Point A (3, 6) is rotated 270° counterclockwise about the origin, what is the

coordinate of A ? Circle the best answer.

(a) 3,6 (c) 3,6

(b) 6,3 (d) 6,3

5. Point J (-2, 5) is rotated 90° counterclockwise about the origin.

What is the coordinate of J′?

(a) 5,2 (c) 2,5

(b) 2,5 (d) 2,5

6. Point D (1, 3) is rotated 180° about the origin. What is the coordinate of D′?

(a) 3,1 (c) 3,1

(b) 3,1 (d) 3,1

7. What single transformation is equivalent to a reflection in the x-axis, followed by

a reflection in the y-axis?


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8. Draw the final image created by rotating polygon ABCD 90° counterclockwise

about the origin and then reflecting the image in the x-axis.

Is the resulting image similar or congruent? How do we know?

9. Figure 1 can be transformed to create Figure 2 using a single transformation.

Which transformation can be used to accomplish this?

A Dilation B Rotation

C Reflection D Translation


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After a dilation occurs are the two figures congruent?

Key Concepts:

Dilation - transformation that produces an image that is the ___________________

as the original but ____________________________.

A dilation is ____________________ to the original figure.

Dilations are centered around the origin (0, 0), unless otherwise stated.

Scale factor: length image-pre

length image, which is a ___________________.

If the scale factor is greater than 1, the figure is

an ____________________________________.

If the scale factor is between 0 and 1, the figure is a

_________________________________________.

Similar Figures- Two figures are said to be similar if their ______________are

congruent and their ____________________________ are proportional.


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1 2 3 4 5 6 7 8–1–2–3–4–5–6–7–8 x

1

2

3

4

5

6

7

8

–1

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–5

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–8

y

Practice:

1. Triangle ABC has vertices A (0, 2), B (4, 4), and C (-1, 4).

What are the vertices of its image with a scale factor of 2?

A’ ( )

B’

C’

What is the rule for this dilation?

,),( yx

Graph the triangle and its image.

2. Quadrilateral ABCD has vertices A (-6, 3), B (3, 3), C (3, -3), and D(- 6, - 6).

It is dilated by a scale factor of 1/3

What are the coordinates of the image?

Graph the quadrilateral and its image.


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3. Which of the following describes the image of a figure after a dilation that has a

scale factor between zero and one?

a) It has a different shape from the original figure and is smaller than the

original figure.

b) It has the same shape as the original and is larger than the original

figure.

c) It has the same shape as the original and is smaller than the original

figure.

d) It has the same shape and same size as the original figure.

4. Which of the following describes the image of a square after a dilation that has a

scale factor of 6?

a) Its sides are 6 units longer than those of the original square.

b) Its sides are 6

1 as long as those of the original square.

c) Its sides are 6 times as long as those of the original square.

d) Its sides are 6 units shorter than those of the original square.

5. Which of the following describes the image of a triangle after a dilation that has

a scale factor of 6

5?

a) Each angle has 6

5 of the measure of its corresponding angle in the

original triangle.

b) Each angle has 5

6 of the measure of its corresponding angle in the

original triangle.

c) Each angle has the same measure as its corresponding angle in the

original triangle.

d) Each angle is 6

1 larger than the measure of its corresponding angle in the

original triangle.


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6. What are the coordinates of PQR after a dilation with a scale factor of 3

2?

a) 2,2,2,0,1,2 RQP b) 4,4,4,0,2,4 RQP

c) 2,4,0,4,2,4 RQP d) 12,12,12,0,6,12 RQP

7. FED is the image of DEF after a dilation with a scale factor of 2. What are

the coordinates of the vertices of DEF ?

a) 4,4,4,8,12,8 FED b) 4,4,0,2,4,6 FED

c) 0,0,4,6,8,2 FED d) 1,1,1,2,3,2 FED


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8. In the diagram below, ∆ 𝐴𝐵𝐶 is similar to ∆𝐴𝑅𝑇.

Part A: What is the scale factor from ∆ 𝐴𝐵𝐶

to ∆ 𝐴𝑅𝑇?

Part B: If the slope of AC is -2, what is the

value of x for coordinate C?

Part C: Using the information from parts A

and B, what is the length of RT?

9. In the coordinate plane below, ∆ 𝐴𝐵𝐶 is similar to ∆ 𝐴𝐸𝐹.

What is the value of x?


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X

Y

Z

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

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–4

–5

–6

y

Practice:

1. If a scale factor is 2

5, how would you write the general rule?

Is this an enlargement or a reduction?

2. Quadrilateral A’B’C’D’ is a dilation of quadrilateral ABCD. Find the scale factor.

Classify the dilation as an enlargement or a reduction.

3. Triangle XYZ is graphed below.

a.) Draw and label Triangle X’Y’Z’ after a dilation using a scale factor of two.

,'),( XX

Y

Z

b.) Are the two triangles similar?

Explain how you know.


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1 2 3 4 5 6 7 8–1–2–3–4–5–6–7–8 x

1

2

3

4

5

6

7

8

–1

–2

–3

–4

–5

–6

–7

–8

y

4. Quadrilateral PQRS has vertices P (-2, 4), Q (4, 4), R (4, -2), and

S (- 4, - 4). It is dilated by a scale factor of ½.

What are the coordinates of the image?

Graph the quadrilateral and its image.

Are the two quadrilaterals similar? Explain.

5. Triangle XYZ has vertices X (0, -2), Y (-1, 2), and Z (2, 2).

What are the vertices of its image with a scale factor of 3?

X’ ( )

Y’

Z’

What is the rule for this dilation?

Graph the triangle and its image.

Are the two triangles similar? Explain.


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6. Triangle PQR has coordinates 6,0,4,2,4,2 RQP . Write the coordinates of

the vertices of the image of a triangle after a dilation of 1.5.

7. How does the size of an image compare to the original figure when the original

figure undergoes a dilation with a scale factor of one?

8. On the grid below, draw the image of FGH after a dilation with a scale factor

of 2

1.

What will be the coordinates of point F” after a translation of polygon F’G’H’

two units to the left and four units up?

Answer ______________


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9. Which transformations produce images that are congruent to their pre-image?

10. Which transformation does not produce an image that is congruent to its pre-

image?

11. a.) What are the two criteria for two figures to be similar?

b.) What are the two criteria for two congruent figures?

12. Which transformations produce figures that have corresponding angles that are

congruent?

13. The table below shows the coordinates of triangle RUN and the coordinates of

R in triangle NUR . Triangle NUR is a dilation of triangle RUN. (4 pts) (8.G.1)

Part A

Fill in the table above for the coordinates of

point U and point N .

Part B

On the graph below, draw and label triangle

NUR after a translation of NUR using the rule

5,2, yxyx .

Part C

Which part(s) of the resulting figure are congruent to the original?

Triangle RUN

Triangle

NUR

R (6, 4) R (3, 2)

U (-8, 0) U

N (2, -2) N


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14. John rotated the figure below 180° about the origin. (6pts) (8.G.4)

John then dilated the figure with a scale factor of ½ with the center of dilation at

the origin.

a. Graph and label the final image on the grid provided.

b. Are the image and the original figure congruent to each other, similar to each

other, or neither? Explain how you determined your answer.


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15. Describe how you could move the solid polygon to exactly match the dashed

polygon using a series of two transformations. (3pts) (8.G.4)

16. What sequence of transformations takes pentagon P to pentagon Q?


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17. Figure Q was the result of a sequence of transformations on Figure P both

shown below.

Which sequence of transformations could take Figure P to Figure Q?

a. Reflection over the x-axis and

translation 7 units right

b. Reflection over the y-axis and

translation 3 units down

c. Translation 1 unit right and 180o

rotation about the origin

d. Translation 4 units right and 180o

rotation about the origin

18. Figure L and Figure M are shown on the grid.

Maria wants to transform figure L to figure M

using only rotations, reflections, and

translations. Which statement is true?

a. The transformation can be done with a

reflection followed by a rotation.

b. The transformation can be done with a

reflection followed by a translation.

c. The transformation cannot be done because

figure L is not congruent to figure M.

d. The transformation cannot be done because figures L and M and are in

different quadrants.


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19. Congruent rectangles HJKL and LKJH are shown on the coordinate grid

below.

Describe a sequence of transformations on

rectangle HJKL that would result in

rectangle LKJH .

20. A series of transformations on quadrilateral S resulted in quadrilateral T.

The angle measures of quadrilateral T are congruent to those of

quadrilateral S.

The side lengths of quadrilateral T are twice as long as those

quadrilateral S.

Which transformation on quadrilateral S must be included to result in

quadrilateral T?

A dilation B reflection

C rotation D translation

21. A triangle with vertices at 1,1A , 1,2B , and 4,1C is translated. The

image of vertex A has coordinates at (3, –1).

Determine the coordinates of either the image of vertex B or the image of vertex C.

Show your work.


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22. Rectangle JKLM is shown on the coordinate

grid.

Rectangle JKLM undergoes a sequence of

transformations, resulting in rectangle J’K’L’M’.

The length of the side K’L’ is 6 units. The

coordinates of vertex K’ are (-3,2), and the

coordinates of vertex M’ are (3, -2).

Describe a sequence of transformations to

rectangle JKLM that would result in rectangle J’K’L’M’. Show your work.

23. Figure X and Figure Y are shown on the

coordinate grid below. Which statement about figures

X and Y must be true.

A. A series of translations will transform

figure X to figure Y, and the figures will be

congruent.

B. A 180⁰ clockwise rotation will transform

figure X to figure Y, and the figures will be

congruent.

C. A series of translations will transform figure X to figure Y, but the figures

will not be congruent.

D. A 180⁰ clockwise rotation will transform figure X to figure Y, but the

figures will not be congruent.


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24. When ABC was dilated by a scale factor of 2, centered at the origin, the result

was its image A’B’C’ shown on the coordinate plane below. The vertices of A’B’C’

are A’(-4, 4), B’(-4, 6), and C’(2, 4).

What are the coordinates of the vertices of ABC?

Vertices A(___, ___) B(___, ___) C(___, ___)

Explain how you determined the coordinates of the vertices of ABC.

Are ABC and A’B’C’ congruent to each other, similar to each other, or neither?

Explain how you determined your answer.


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25. Quadrilateral ABCD is graphed on a coordinate plane.

Abby reflected ABCD over the x-axis and then rotated it 90° clockwise about

the origin. She labeled the final image EFGH.

Manny dilated ABCD by a scale factor of 3 and then translated the resulting

figure 2 units left. He labeled the final image PQRS.

Identify a pair of quadrilaterals from the three quadrilaterals ABCD, EFGH,

and PQRS that are congruent.

Answer ____________________________________

Identify a pair of quadrilaterals from the three quadrilaterals ABCD, EFGH,

and PQRS that are similar but not congruent.

Answer ____________________________________

Describe a transformation on Abby’s quadrilateral EFGH that would make

the resulting image E’F’G’H’ congruent to Manny’s quadrilateral PQRS.

26. Rectangle FGHJ shown below, is translated 6 units right and 1 unit up to

produce rectangle F’G’H’J’.

Which statement about the side lengths of

rectangle F’G’H’J’ is true?

A F’G’ = 3 and G’H’ = 5

B F’G’ = 3 and G’H’ = 6

C F’G’ = 9 and G’H’ = 5

D F’G’ = 9 and G’H’ = 6


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27. Mia enlarged a plan for an outdoor stage. The original plan is shown below.

She dilated the outdoor stage by a scale

factor of four with the center of dilation at

the origin. Which ordered pair will be the

coordinates of one of the new vertices?

A (2,1)

B (8,16)

C (32, 4)

D (32, 16)

28. Part A Fill in the table below for the coordinates of H , K , and L after a reflection

over the x-axis. (4 pts)(8.G.3)

Part B: On the graph below, draw and label triangle HKL

and triangle LKH .

Part C: Explain how the coordinates of H

changed to the coordinates of H .

Triangle

HKL

Triangle

LKH

H (-2, 3) H

K (4, 2) K

L (3, -2) L


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29. When ABC was translated using the rule 3,5),( yxyx , the result was

its image. CBA shown on the coordinate plane below.

The vertices of CBA are 4,4A , 6,4B , and 4,2C . (4pts) (8.G.2)

What are the coordinates of the vertices of ABC ? Explain how you determined

them.

What is the length of CA . How does this compare with the length of AC ? Why?

30. Triangle M is similar to triangle N. Triangle M has two angles with measures of

32° and 93°. Which two angle measures could be included in triangle N?

A 32° and 58°

B 32° and 74°

C 93° and 55°

D 93° and 87°


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1. a. In the figure below, 𝐿1 is not parallel to 𝐿2, and 𝑚 is a transversal. Use patty paper to trace angles 1–4. Which, if any, are equal? Explain why.

b. What other pairs of angles would be congruent based on question 1?

c. Now translate angle 1-4 down to see if any of these angles are congruent to angles 5-8. What do you notice?

d. Given that the 𝑚 < 1 = 115° and the 𝑚 < 2 = 65°, what other angles can you determine?

e. What do you notice about the sum of angle 1 and angle 2?

f. What do you notice about the sum of angle 1, 2, 3, and 4?


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2. In the figure below, 𝐿1 ∥ 𝐿2, and 𝑚 is a transversal.

a. If the 𝑚 < 1 = 40°, what other angle measures do we know based on problem 1?

b. Use patty paper to trace angles 1-4. What did you notice about the measures of ∠1 and ∠5? What transformation explains this?

c. What do you notice about the measures of ∠3 and ∠7? What transformation explains this? Are there any other pairs of angles with this same relationship? If so, list them.

Angles that are on the __________________ side of the transversal in the same position are called ____________________________________________.

If 𝐿1 and 𝐿2 are __________________________, then the corresponding angles are ______________________________.


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d. What do you notice about the measures of ∠4 and ∠6? What transformations explain this? Is there another pair of angles with this same relationship?

When angles are on _________________________ sides of the transversal and

inside the lines 𝐿1 and 𝐿2, they are called ___________________________________________.

If 𝐿1 and 𝐿2 are __________________________, then the alternate interior angles are

______________________________.

e. What do you notice about the measures of ∠2 and ∠8? What transformations explain this? Is there another pair of angles with this same relationship?

When angles are on _________________________ sides of the transversal and outside of the

parallel lines, they are called ___________________________________ angles.

If 𝐿1 and 𝐿2 are __________________________, then the alternate exterior angles are

______________________________.


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f. Based on the different relationships described, determine the measures of all the

angles in the diagram. What do you notice about the measures of ∠4 and ∠5? What do you notice about the measures of ∠3 and ∠6?

Angles that sum to _________ are called ___________________________ angles.

When angles are on the _________________ side of the transversal and inside of the

parallel lines, they are called ___________________________________________ angles.

If 𝐿1 and 𝐿2 are __________________________, then the same side interior angles are

______________________________________.


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Exterior Angles of a Triangle

5. How can you calculate the measure of an exterior angle of a triangle?

6. How does that help you find the measure of the missing angle in the triangle below?


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Practice:

1. In the diagram below, lines L1 and L2 are parallel. What are the measures of

angles a–g?

2. What are the measures of angles 𝑎 and 𝑏 in the triangle at the right?

3. What is the value of 𝑥 in the diagram at the right?

4. What is the value of n in the diagram below?

A 18

B 24

C 42

D 48


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Find the value of the variable. Show your work or explain.

10. 11.


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1. a. In the following diagram m ∥ l cut by transversal t, find the measure of the

other eight angles if m∠4 = 110°.

m∠1 = ________ m∠2 = ________ m∠3 = ________

m∠5 = ________ m∠6 = ________ m∠7 = ________

m∠8 = ________

b. Identify a pair of corresponding angles:

c. Identify a pair of alternate interior angles:

d. Identify a pair of alternate exterior angles:


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2. In the diagram below lines l and k are parallel.

Part A: What is the value of x? Show work.

Part B: What is the measure, in degrees, of < 𝐴𝐵𝐶


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3. If m∠3 = 90° and m∠CBD = 35°.

Determine the measures of the remaining angles in the diagram.

Write the measures in the diagram.

Explain how you determined each of them.

Color congruent angles the same color.

4. Find the value of x. Show work. 5. Find the value of x. Show

work and explain.


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6. If m∠3 = 45° and m∠DCA = 135°.

Determine the measures of the remaining angles in the diagram.

Write the measures in the diagram.

Explain how you determined each of them.

Color congruent angles the same color.

7. If l || m. and m∠1 = 50°, determine the measures of the remaining angles.

Give an example of alternate exterior angles.


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8. If l || m. and m∠2 = 110°, determine the measures of the remaining angles.

9. The m∠2 = 90°. Find the measure of the remaining angles if m∠4 = 30°.

m∠1 = ________ m∠2 = ________

m∠3 = ________ m∠5 = ________

Tell how you know their measure without the use of a protractor.

10. Find the degree measure of YXZ .

1

23

4

5


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11. If the 533m and 854m find all the angles.

Explain how you determined 7m .

12. In the following diagram a║ b.is cut by a transversal.

The 1m is 118o. Find the 4m and explain how you determined the

measurement.

13. In parallelogram ABCD, the Am is 47o. What is the measure of B ? Explain.


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14. In ΔABC, the m B is twice the m A, and the m C is three times the m A.

Find the number of degrees in each angle of the triangle. Show work.

15. Given rectangle ACDF, 20BADm and 60CBDm .

Find the measurement of ADE .

Explain how you arrived at the measurement for ADE .

16. If the 405xWXYm , 70YXZm and 142xXYZm find XZYm .


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17. In the following diagram m║l cut by transversal t1 and t2.

Find the measure of the angles 1, 3, & 5 and provide reasoning.

Angle Measure Reason

1

3

5

18. In the following diagram n1║ n2.is cut by a transversal.

The 8m is 65o. Find the 2m and explain how you determined the measurement.


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29. If the 731 xm , 1711 xm and 1108m find 4m .

30. In the following diagram x║ y.is cut by a transversal.

The 1m is 102o. Find the 3m and explain how you determined the measurement.


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31. Find the value of x. Show work and explain.

32. Find the PRQm . Show work.

33. Two complementary angles measure 53 x and 75 x .

What is the measure of the larger angle? Show work.

34. Find the value of y. Find the degree measure of ABC .

-6y+52

9y+7

A

B C

D E

F


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35.

Explain why.

36. 37.


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