Problem 1 Got It? Objective To identify congruence transformations To prove triangle congruence using isometries Congruence Transformations Suppose that you want to create two identical wings for a model airplane. You draw one wing on a large sheet of tracing paper, fold it along the dashed line, and then trace the first wing. How do you know that the two wings are identical? In the Solve It, you may have used the properties of rigid motions to describe why the wings are identical. Essential Understanding You can use compositions of rigid motions to understand congruence. Are there other methods you could use to create two identical wings? Lesson Vocabulary • congruent • congruence transformation L V Identifying Equal Measures e composition (r (90°, P) ∘ R n )(LMNO) = GHJK is shown at the right. A Which angle pairs have equal measures? Because compositions of isometries preserve angle measure, corresponding angles have equal measures. m∠L = m∠G, m∠M = m∠H, m∠N = m∠J , and m∠O = m∠K . B Which sides have equal lengths? By definition, isometries preserve distance. So, corresponding side lengths have equal measures. LM = GH, MN = HJ, NO = JK, and LO = GK. 1. e composition (R t ∘ T 62, 37 )(△ABC) = △XYZ. List all of the pairs of angles and sides with equal measures. n P H G J K L M N O How can you use the properties of isometries to find equal angle measures and equal side lengths? Isometries preserve angle measure and distance, so identify corresponding angles and corresponding side lengths. CC-13 MACC.912.G-CO.2.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent . . . Also MACC.912.G-CO.2.6, MACC.912.G-CO.2.8 MP 1, MP 3, MP 4 MATHEMATICAL PRACTICES Common Core State Standards 54 Common Core 54 Common Core BIG ideas Transformations Visualization ESSENTIAL UNDERSTANDING • If two figures can be mapped to each other by a sequence of rigid motions, then the figures are congruent. Math Background In previous lessons, students have learned that two figures are congruent if and only if corresponding sides have the same length and corresponding angles have the same measure. In this lesson, students explore the concept of congruence transformations. Two figures are congruent if and only if there is a rigid motion or composition of rigid motions that maps one figure to the other. This new approach to determining congruence can be used to verify postulates such as the SAS Postulate or the SSS Postulate. Students will learn that if there is a way to map one figure onto another through a series of rigid motions, then the figures are congruent. Mathematical Practices Construct viable arguments and critique the reasoning of others. Students will use the properties of rigid motions to construct arguments for the validity of the SAS and SSS congruence postulates. Preparing to Teach 1 Interactive Learning Solve It! PURPOSE To develop an intuitive sense of congruence when working with rigid motions PROCESS Students may determine that the two wings are identical because one was formed by tracing the other so they overlap exactly. Problem 1 In this problem, students will identify corresponding angles and side lengths after a composition of rigid motions. Got It? FACILITATE Q Which angles do you think have the same measures in the two figures? Explain. [corresponding angles; the figures are congruent] Q Which sides do you think have the same lengths in the two figures? Explain. [corresponding sides; the figures are congruent] Q Would the answer change if trapezoid LMNO were rotated first and then reflected to create trapezoid GHJK? Explain. [No, the corresponding angles and sides would still be the same.] Q How can you use the order of the vertices of the preimage and image to solve the problem? [The corresponding vertices are listed in order in the composition statement.] CC-13
8
Embed
Common Core State Standards 1 Interactive Learning CC-13 ...€¦ · Lesson 9-5 Congruence Transformations 55 In Problem 1 you saw that compositions of rigid motions preserve corresponding
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Problem 1
Got It?
54 Chapter 9 Transformations
Objective To identify congruence transformations To prove triangle congruence using isometries
Congruence Transformations
Suppose that you want to create two identical wings for a model airplane. You draw one wing on a large sheet of tracing paper, fold it along the dashed line, and then trace the first wing. How do you know that the two wings are identical?
In the Solve It, you may have used the properties of rigid motions to describe why the wings are identical.
Essential Understanding You can use compositions of rigid motions to understand congruence.
Are there other methods you could use to create two identical wings?
Lesson Vocabulary
•congruent•congruence
transformation
LessonVocabulary
Identifying Equal Measures
The composition (r(90°, P) ∘ Rn)(LMNO) = GHJK is shown at the right.
A Which angle pairs have equal measures?
Because compositions of isometries preserve angle measure, corresponding angles have equal measures.
By definition, isometries preserve distance. So, corresponding side lengths have equal measures.
LM = GH, MN = HJ, NO = JK, and LO = GK.
1. The composition (Rt ∘ T62, 37)(△ABC) = △XYZ . List all of the pairs of angles and sides with equal measures.
geom12_se_ccs_c09l05_t02.ai
n
P HG
J
K
L M
NO
How can you use the properties of isometries to find equal angle measures and equal side lengths? Isometries preserve angle measure and distance, so identify corresponding angles and corresponding side lengths.
CC-13 MACC.912.G-CO.2.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent . . . Also MACC.912.G-CO.2.6, MACC.912.G-CO.2.8
In Problem 1 you saw that compositions of rigid motions preserve corresponding side lengths and angle measures. This suggests another way to define congruence.
Identifying Congruent Figures
Which pairs of figures in the grid are congruent? For each pair, what is a sequence of rigid motions that maps one figure to the other?
Figures are congruent if and only if there is a sequence of rigid motions that maps one figure to the other. So, to find congruent figures, look for sequences of translations, rotations, and reflections that map one figure to another.
Because r(180°, O)(△DEF) = △LMN , the triangles are congruent.
Because (T6-1, 57 ∘ Ry@axis)(ABCJ) = WXYZ , the trapezoids are congruent.
Because T6-2, 97(HG) = PQ, the line segments are congruent.
2. Which pairs of figures in the grid are congruent? For each pair, what is a sequence of rigid motions that map one figure to the other?
Key Concept Congruent Figures
Two figures are congruent if and only if there is a sequence of one or more rigid motions that maps one figure onto the other.
geom12_se_ccs_c09l05_t03_updated.ai
y
xO2 2
2
6 6
6
4
2
6
4
4
HG
D
E
F
A B
CJ
L
M
N
Y
X WQ
P
Z
geom12_se_ccs_c09l05_t04_updated.ai
y
xO 2
2
6 6
6
4
2
6
4
4
4
F
B
A
C
D UV
W
H K
I JQ
N
M
Does one rigid motion count as a sequence? Yes. It is a sequence of length 1.
Math BackgroundInpreviouslessons,studentshavelearnedthattwofiguresarecongruentifandonlyifcorrespondingsideshavethesamelengthandcorrespondingangleshavethesamemeasure.Inthislesson,studentsexploretheconceptofcongruencetransformations.Twofiguresarecongruentifandonlyifthereisarigidmotionorcompositionofrigidmotionsthatmapsonefiguretotheother.
Mathematical PracticesConstruct viable arguments and critique the reasoning of others. StudentswillusethepropertiesofrigidmotionstoconstructargumentsforthevalidityoftheSASandSSScongruencepostulates.
Problem 1 Inthisproblem,studentswillidentifycorrespondinganglesandsidelengthsafteracompositionofrigidmotions.
Got It?
FACILITATEQ Whichanglesdoyouthinkhavethesame
measuresinthetwofigures?Explain. [corresponding angles; the figures are congruent]
Q Whichsidesdoyouthinkhavethesamelengthsinthetwofigures?Explain. [corresponding sides; the figures are congruent]
Q WouldtheanswerchangeiftrapezoidLMNOwererotatedfirstandthenreflectedtocreatetrapezoidGHJK?Explain. [No, the corresponding angles and sides would still be the same.]
Q Howcanyouusetheorderoftheverticesofthepreimageandimagetosolvetheproblem? [The corresponding vertices are listed in order in thecompositionstatement.]
CC-13
Problem 1
Got It?
54 Chapter 9 Transformations
Objective To identify congruence transformations To prove triangle congruence using isometries
Congruence Transformations
Suppose that you want to create two identical wings for a model airplane. You draw one wing on a large sheet of tracing paper, fold it along the dashed line, and then trace the first wing. How do you know that the two wings are identical?
In the Solve It, you may have used the properties of rigid motions to describe why the wings are identical.
Essential Understanding You can use compositions of rigid motions to understand congruence.
Are there other methods you could use to create two identical wings?
Lesson Vocabulary
•congruent•congruence
transformation
LessonVocabulary
Identifying Equal Measures
The composition (r(90°, P) ∘ Rn)(LMNO) = GHJK is shown at the right.
A Which angle pairs have equal measures?
Because compositions of isometries preserve angle measure, corresponding angles have equal measures.
By definition, isometries preserve distance. So, corresponding side lengths have equal measures.
LM = GH, MN = HJ, NO = JK, and LO = GK.
1. The composition (Rt ∘ T62, 37)(△ABC) = △XYZ . List all of the pairs of angles and sides with equal measures.
geom12_se_ccs_c09l05_t02.ai
n
P HG
J
K
L M
NO
How can you use the properties of isometries to find equal angle measures and equal side lengths? Isometries preserve angle measure and distance, so identify corresponding angles and corresponding side lengths.
CC-13 MACC.912.G-CO.2.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent . . . Also MACC.912.G-CO.2.6, MACC.912.G-CO.2.8
In Problem 1 you saw that compositions of rigid motions preserve corresponding side lengths and angle measures. This suggests another way to define congruence.
Identifying Congruent Figures
Which pairs of figures in the grid are congruent? For each pair, what is a sequence of rigid motions that maps one figure to the other?
Figures are congruent if and only if there is a sequence of rigid motions that maps one figure to the other. So, to find congruent figures, look for sequences of translations, rotations, and reflections that map one figure to another.
Because r(180°, O)(△DEF) = △LMN , the triangles are congruent.
Because (T6-1, 57 ∘ Ry@axis)(ABCJ) = WXYZ , the trapezoids are congruent.
Because T6-2, 97(HG) = PQ, the line segments are congruent.
2. Which pairs of figures in the grid are congruent? For each pair, what is a sequence of rigid motions that map one figure to the other?
Key Concept Congruent Figures
Two figures are congruent if and only if there is a sequence of one or more rigid motions that maps one figure onto the other.
geom12_se_ccs_c09l05_t03_updated.ai
y
xO2 2
2
6 6
6
4
2
6
4
4
HG
D
E
F
A B
CJ
L
M
N
Y
X WQ
P
Z
geom12_se_ccs_c09l05_t04_updated.ai
y
xO 2
2
6 6
6
4
2
6
4
4
4
F
B
A
C
D UV
W
H K
I JQ
N
M
Does one rigid motion count as a sequence? Yes. It is a sequence of length 1.
Problem 2 VisUal lEarnErsPointouttostudentsthatthefirststepshouldalwaysbetolookfortwofiguresthatappeartohavethesamesizeandshape.Thentoverifycongruence,theymustidentifyarigidmotionorcompositionofrigidmotionsthatmapsonefiguretotheother.
Got It?
Q Isthereanothercompositionofrigidmotionsthatyoucouldhaveusedtodeterminethecongruenceof△DEF and△LMN?Explain. [Yes. For example, a composition of reflections across both axes.]
Q Howmightyouhaveverifiedthecongruenceof △UVWand△MNOpriortolearningthemethodsofthislesson?Explain. [Sample answer: Use the Distance Formula to show that the sides have the same lengths and then apply the SSS Postulate.]
Problem 3
Got It?
56 Chapter 9 Transformations
Because compositions of rigid motions take figures to congruent figures, they are also called congruence transformations.
Identifying Congruence Transformations
In the diagram at the right, △JQV @ △EWT. What is a congruence transformation that maps △JQV onto △EWT ?
Because △EWT lies above △JQV on the plane, a translation can map △JQV up on the plane. Also, notice that △EWT is on the opposite side of the y-axis and has the opposite orientation of △JQV. This suggests that the triangle is reflected across the y-axis.
It appears that a translation of △JQV up 5 units, followed by a reflection across the y-axis maps △JQV to △EWT. Verify by using the coordinates of the vertices.
T60, 57(x, y) = (x, y + 5) T60, 57(J) = (2, 4)
Ry@axis(2, 4) = (-2, 4) = E
Next, verify that the sequence maps Q to W and V to T.
So, the congruence transformation Ry@axis ∘ T60, 57 maps △JQV onto △EWT . Note that there are other possible congruence transformations that map △JQV onto △EWT .
3. What is a congruence transformation that maps △NAV to △BCY ?
geom12_se_ccs_c09l05_t05.ai
y
x
T
E
Q
J
V
WO
2
4
2
4
42
4
Identify the corresponding parts and find a congruence transformation that maps the preimage to the image. Then use the vertices to verify the congruence transformation.
A sequence of rigid motions that maps △JQV onto △EWT
Earlier in the course, you studied triangle congruence postulates and theorems. You can use congruence transformations to justify criteria for determining triangle congruence.
Verifying the SAS Postulate
Given: ∠J ≅ ∠P, PA ≅ JO, FP ≅ SJ
Prove: △JOS ≅ △PAF
Step 1 Translate △PAF so that points A and O coincide.
Step 2 Because PA ≅ JO, you can rotate △PAF about point A so that PA and JO coincide.
Step 3 Reflect △PAF across PA. Because reflections preserve angle measure and distance, and because ∠J ≅ ∠P and FP ≅ SJ , you know that the reflection maps ∠P to ∠J and FP to SJ . Since points S and F coincide, △PAF coincides with △JOS.
There is a congruence transformation that maps △PAF onto △JOS, so △PAF ≅ △JOS.
4. Verify the SSS postulate.
Given: TD ≅ EN , YT ≅ SE, YD ≅ SN
Prove: △YDT ≅ △SNE
Proof
geom12_se_ccs_c09l05_t08.ai
S
O
J
P
F A
geom12_se_ccs_c09l05_t09.ai
O
S
J
A
P
F
geom12_se_ccs_c09l05_t010.ai
P
A
F
O
J
S
geom12_se_ccs_c09l05_t011.ai
OA
FS
PJ
geom12_se_ccs_c09l05_t012.ai
T
D
E
S
N
Y
In Problem 4, you used the transformational approach to prove triangle congruence. Because this approach is more general, you can use what you know about congruence transformations to determine whether any two figures are congruent.
How do you show that the two triangles are congruent?Find a congruence transformation that maps one onto the other.
Problem 3 Inthisproblem,studentswillidentifyacongruencetransformationthatmapsonetriangleontoanother.
Got It?
Q Isthereasinglerigidmotionthatcanbeusedtomap△JQV onto△EW T?Explain. [No, △JQVcannot be reflected, translated, or rotated to overlap △EW T . A composition of rigid motions is necessary.]
Q Whatdoyouneedtoshowtoprovethatthetrianglesarecongruent? [You need to show that there is a congruence transformation from one triangle onto the other]
Q Lookingattheorientationofthetriangles,howdoyouthinkonetrianglecanbetransformedtomapittotheothertriangle? [Sample answer: translation and rotation]
Because compositions of rigid motions take figures to congruent figures, they are also called congruence transformations.
Identifying Congruence Transformations
In the diagram at the right, △JQV @ △EWT. What is a congruence transformation that maps △JQV onto △EWT ?
Because △EWT lies above △JQV on the plane, a translation can map △JQV up on the plane. Also, notice that △EWT is on the opposite side of the y-axis and has the opposite orientation of △JQV. This suggests that the triangle is reflected across the y-axis.
It appears that a translation of △JQV up 5 units, followed by a reflection across the y-axis maps △JQV to △EWT. Verify by using the coordinates of the vertices.
T60, 57(x, y) = (x, y + 5) T60, 57(J) = (2, 4)
Ry@axis(2, 4) = (-2, 4) = E
Next, verify that the sequence maps Q to W and V to T.
So, the congruence transformation Ry@axis ∘ T60, 57 maps △JQV onto △EWT . Note that there are other possible congruence transformations that map △JQV onto △EWT .
3. What is a congruence transformation that maps △NAV to △BCY ?
geom12_se_ccs_c09l05_t05.ai
y
x
T
E
Q
J
V
WO
2
4
2
4
42
4
Identify the corresponding parts and find a congruence transformation that maps the preimage to the image. Then use the vertices to verify the congruence transformation.
A sequence of rigid motions that maps △JQV onto △EWT
Earlier in the course, you studied triangle congruence postulates and theorems. You can use congruence transformations to justify criteria for determining triangle congruence.
Verifying the SAS Postulate
Given: ∠J ≅ ∠P, PA ≅ JO, FP ≅ SJ
Prove: △JOS ≅ △PAF
Step 1 Translate △PAF so that points A and O coincide.
Step 2 Because PA ≅ JO, you can rotate △PAF about point A so that PA and JO coincide.
Step 3 Reflect △PAF across PA. Because reflections preserve angle measure and distance, and because ∠J ≅ ∠P and FP ≅ SJ , you know that the reflection maps ∠P to ∠J and FP to SJ . Since points S and F coincide, △PAF coincides with △JOS.
There is a congruence transformation that maps △PAF onto △JOS, so △PAF ≅ △JOS.
4. Verify the SSS postulate.
Given: TD ≅ EN , YT ≅ SE, YD ≅ SN
Prove: △YDT ≅ △SNE
Proof
geom12_se_ccs_c09l05_t08.ai
S
O
J
P
F A
geom12_se_ccs_c09l05_t09.ai
O
S
J
A
P
F
geom12_se_ccs_c09l05_t010.ai
P
A
F
O
J
S
geom12_se_ccs_c09l05_t011.ai
OA
FS
PJ
geom12_se_ccs_c09l05_t012.ai
T
D
E
S
N
Y
In Problem 4, you used the transformational approach to prove triangle congruence. Because this approach is more general, you can use what you know about congruence transformations to determine whether any two figures are congruent.
How do you show that the two triangles are congruent?Find a congruence transformation that maps one onto the other.
Problem 4 EXtEnsiOnInthisproblem,studentswillverifytheSASPostulateforprovingtrianglecongruencebyfindingacongruencetransformationthatmapsonetriangleontotheother.
Got It?
Q DoyouthinkyoucouldusesimilarmethodstoverifytheSSSPostulate,theASAPostulate,andtheAASTheorem?Explain. [Yes, the only difference would be the given information. In all cases, you would simply identify a congruence transformation that maps one triangle to the other.]
Q Ifafriendishavingdifficultyseeingthecongruencetransformationinthisproblem,howmightyouhelphimorher? [Sample answer: Suggest tracing the triangles on tracing paper and cutting them out. Then slide, flip, and turn the cut outs until they overlap.]
Problem 5
Got It?
58 Chapter 9 Transformations
Determining Congruence
Is Figure A congruent to Figure B? Explain how you know.
Figure A can be mapped to Figure B by a sequence of reflections or a simple translation. So, Figure A is congruent to Figure B because there is a congruence transformation that maps one to the other.
5. Are the figures shown at the right congruent? Explain.
Figure A
Figure B
geom12_se_ccs_c09l05_t014.ai
Do you know HOW?Use the graph for Exercises 1 and 2.
1. Identify a pair of congruent figures and write a congruence statement.
2. What is a congruence transformation that relates two congruent figures?
Do you UNDERSTAND? 3. How can the definition of congruence in terms of
rigid motions be more useful than a definition of congruence that relies on corresponding angles and sides?
4. Reasoning Is a composition of a rotation followed by a glide reflection a congruence transformation? Explain.
5. Open Ended What is an example of a board game in which a game piece is moved by using a congruence transformation?
geom12_se_ccs_c09l05_t015.ai
R
x
y
V
A
T B
K
QS
I
O
2
6 4 2
2
6
4
4
Lesson Check
Practice and Problem-Solving Exercises
For each coordinate grid, identify a pair of congruent figures. Then determine a congruence transformation that maps the preimage to the congruent image.
6. 7. 8.
PracticeA See Problem 1 and 2.
geom12_se_ccs_c09l05_t016.ai
x
y
B
J
T
V
Q
E
YL
G
2 44
4
4
geom12_se_ccs_c09l05_t017.ai
G
A
D
F
C
R
y
xO4 2
2
4
4
4
geom12_se_ccs_c09l05_t018.ai
x
y
FA
E
K
S
T
B
D
M
I
C
2
4
4
4
How can you determine whether the figures are congruent?You can find a congruence transformation that maps Figure A onto Figure B.
In Exercises 9–11, find a congruence transformation that maps △LMN to △RST .
9. 10. 11.
12. Verify the ASA Postulate for triangle congruence by using congruence transformations.
Given: EK ≅ LH Prove: △EKS ≅ △HLA
∠E ≅ ∠H
∠K ≅ ∠L
13. Verify the AAS Postulate for triangle congruence by using congruence transformations.
Given: ∠I ≅ ∠V Prove: △NVZ ≅ △CIQ
∠C ≅ ∠N
QC ≅ NZ
In Exercises 14–16, determine whether the figures are congruent. If so, describe a congruence transformation that maps one to the other. If not, explain.
14. 15. 16.
Construction The figure at the right shows a roof truss of a new building. Identify an isometry or composition of isometries to justify each of the following statements.
3 Lesson CheckDo you know HOW? ErrOr intErVEntiOn•IfstudentshavetroublesolvingExercise1,thenaskthemwhichtrianglescouldbeplacedontopofeachothersothattheycoincide.
Do you UNDERSTAND?•HavestudentssharetheirresponsestoExercise5 withtherestoftheclasssothateveryoneisexposedtodifferentrealworldexamplesofcongruencetransformations.
Close
Q Howcanyoushowthattwofiguresarenotcongruent? [If corresponding distances are not equal, figures are not congruent.]
Q Whatdoyouknowaboutcorrespondingsidesandanglesoffiguresthataremappedusingacompositionofrigidmotions? [They have equal measures.]
Q Howcanyoushowthattwofiguresarecongruent? [Show that there is a sequence of rigid motions that maps one figure onto the other.]
Q Supposetwofiguresarecongruent.Whatdoyouknowabouthowthefiguresarerelatedintheplane? [There is a congruence transformation that maps one figure onto the other.]
Problem 5
Got It?
58 Chapter 9 Transformations
Determining Congruence
Is Figure A congruent to Figure B? Explain how you know.
Figure A can be mapped to Figure B by a sequence of reflections or a simple translation. So, Figure A is congruent to Figure B because there is a congruence transformation that maps one to the other.
5. Are the figures shown at the right congruent? Explain.
Figure A
Figure B
geom12_se_ccs_c09l05_t014.ai
Do you know HOW?Use the graph for Exercises 1 and 2.
1. Identify a pair of congruent figures and write a congruence statement.
2. What is a congruence transformation that relates two congruent figures?
Do you UNDERSTAND? 3. How can the definition of congruence in terms of
rigid motions be more useful than a definition of congruence that relies on corresponding angles and sides?
4. Reasoning Is a composition of a rotation followed by a glide reflection a congruence transformation? Explain.
5. Open Ended What is an example of a board game in which a game piece is moved by using a congruence transformation?
geom12_se_ccs_c09l05_t015.ai
R
x
y
V
A
T B
K
QS
I
O
2
6 4 2
2
6
4
4
Lesson Check
Practice and Problem-Solving Exercises
For each coordinate grid, identify a pair of congruent figures. Then determine a congruence transformation that maps the preimage to the congruent image.
6. 7. 8.
PracticeA See Problem 1 and 2.
geom12_se_ccs_c09l05_t016.ai
x
y
B
J
T
V
Q
E
YL
G
2 44
4
4
geom12_se_ccs_c09l05_t017.ai
G
A
D
F
C
R
y
xO4 2
2
4
4
4
geom12_se_ccs_c09l05_t018.ai
x
y
FA
E
K
S
T
B
D
M
I
C
2
4
4
4
How can you determine whether the figures are congruent?You can find a congruence transformation that maps Figure A onto Figure B.
In Exercises 9–11, find a congruence transformation that maps △LMN to △RST .
9. 10. 11.
12. Verify the ASA Postulate for triangle congruence by using congruence transformations.
Given: EK ≅ LH Prove: △EKS ≅ △HLA
∠E ≅ ∠H
∠K ≅ ∠L
13. Verify the AAS Postulate for triangle congruence by using congruence transformations.
Given: ∠I ≅ ∠V Prove: △NVZ ≅ △CIQ
∠C ≅ ∠N
QC ≅ NZ
In Exercises 14–16, determine whether the figures are congruent. If so, describe a congruence transformation that maps one to the other. If not, explain.
14. 15. 16.
Construction The figure at the right shows a roof truss of a new building. Identify an isometry or composition of isometries to justify each of the following statements.
20. Vocabulary If two figures are ________________, then there is an isometry that maps one figure onto the other.
21. Think About a Plan The figure at the right shows two congruent, isosceles triangles. What are four different isometries that map the top triangle onto the bottom triangle?
• How can you use the three basic rigid motions to map the top triangle onto the bottom triangle?
• What other isometries can you use?
22. Graphic Design Most companies have a logo that is used on company letterhead and signs. A graphic designer sketches the logo at the right. What congruence transformations might she have used to draw this logo?
23. Art Artists frequently use congruence transformations in their work. The artworks shown below are called tessellations. What types of congruence transformations can you identify in the tessellations?
a. b.
24. In the footprints shown below, what congruence transformations can you use to extend the footsteps?
25. Prove the statements in parts (a) and (b) to show congruence in terms of transformations is equivalent to the criteria of for triangle congruence you learned in Chapter 4.
a. If there is a congruence transformation that maps △ABC to △DEF then corresponding pairs of sides and corresponding pairs of angles are congruent.
b. In △ABC and △DEF , if corresponding pairs of sides and corresponding pairs of angles are congruent, then there is a congruence transformation that maps △ABC to △DEF .
26. Baking Cookie makers often use a cookie press so that the cookies all look the same. The baker fills a cookie sheet for baking in the pattern shown. What types of congruence transformations are being used to set each cookie on the sheet?
27. Use congruence transformations to prove the Isosceles Triangle Theorem.
Given: FG ≅ FH
Prove: ∠G ≅ ∠H
28. Reasoning You project an image for viewing in a large classroom. Is the projection of the image an example of a congruence transformation? Explain your reasoning.
22. reflectionorrotation 23a. rotationsandglidereflections b. translationsandglidereflections 24. glidereflection 25a. congruencetransformationspreserve
distancesandanglemeasures b. UseSAS,proveninProblem4.
60 Chapter 9 Transformations
20. Vocabulary If two figures are ________________, then there is an isometry that maps one figure onto the other.
21. Think About a Plan The figure at the right shows two congruent, isosceles triangles. What are four different isometries that map the top triangle onto the bottom triangle?
• How can you use the three basic rigid motions to map the top triangle onto the bottom triangle?
• What other isometries can you use?
22. Graphic Design Most companies have a logo that is used on company letterhead and signs. A graphic designer sketches the logo at the right. What congruence transformations might she have used to draw this logo?
23. Art Artists frequently use congruence transformations in their work. The artworks shown below are called tessellations. What types of congruence transformations can you identify in the tessellations?
a. b.
24. In the footprints shown below, what congruence transformations can you use to extend the footsteps?
25. Prove the statements in parts (a) and (b) to show congruence in terms of transformations is equivalent to the criteria of for triangle congruence you learned in Chapter 4.
a. If there is a congruence transformation that maps △ABC to △DEF then corresponding pairs of sides and corresponding pairs of angles are congruent.
b. In △ABC and △DEF , if corresponding pairs of sides and corresponding pairs of angles are congruent, then there is a congruence transformation that maps △ABC to △DEF .
26. Baking Cookie makers often use a cookie press so that the cookies all look the same. The baker fills a cookie sheet for baking in the pattern shown. What types of congruence transformations are being used to set each cookie on the sheet?
27. Use congruence transformations to prove the Isosceles Triangle Theorem.
Given: FG ≅ FH
Prove: ∠G ≅ ∠H
28. Reasoning You project an image for viewing in a large classroom. Is the projection of the image an example of a congruence transformation? Explain your reasoning.