High School: Congruence and Rigid Motion
CONGRUENCE
AND RIGID MOTION
QUIZ 1 ANSWERS
Total: 11 points
1.
G-CO-2
1 point
B Rotation
2.
G-CO-2
1 point
A Translation
3.
G-CO-2
1 point
C Reflection
4.
G-CO-5
2 points
y
D'
8
F'
6
4
E'
2
E
–6
–4
–2
2
4
x
6
F
–2
2
D
–4
5.
G-CO-5
1 point
Draws the line of reflection correctly.
1 point
Plots 3 points correctly.
4 points
y
I'
H'
I
J
4
J'
G'
–6
–4
2
–2
G
H
2
4
6
–2
1 point
Plots J' at (–4, 3).
1 point
Plots I' at (–5, 6).
1 point
Plots H' at (–1, 5).
1 point
Plots G' at (–2, 2).
Copyright © 2014 Pearson Education, Inc. 1
x
High School: Congruence and Rigid Motion
CONGRUENCE
AND RIGID MOTION
6.
G-CO-5
QUIZ 1 ANSWERS
2 points
y
6
Z
4
Y
–6
–4
2
Z'
X
–2
2
4
6
x
–2
Y'
–4
Copyright © 2014 Pearson Education, Inc. X'
1 point
Translates the x-values correctly.
1 point
Translates the y-values correctly.
2
High School: Congruence and Rigid Motion
CONGRUENCE
AND RIGID MOTION
QUIZ 2 ANSWERS
Total: 10 points
1.
G-CO-6
1 point
C Translation T(x, y) = (x, y – 7)
2.
G-CO-6
1 point
C Rotation of 180° around the origin
3.
G-CO-8
1 point
D AAS
4.
G-CO-8
1 point
A SAS
5.
G-CO-6
3 points
Possible answer:
∆FGH is congruent to ∆JKL because there is a series of rigid motions that link the two
together. You can map the first triangle directly onto the second triangle with two
rigid transformations. First, translate ∆FGH 5 units to the left and then 1 unit down.
This maps point F onto point J. Then rotate the figure around the point (–3, 1) 90°.
This maps points G and H onto the points K and L.
Since there is a series of rigid transformations that maps the first triangle exactly
onto the second, the two triangles must be congruent
6.
G-CO-8
3 points
1 point
Uses the logic that rigid motions imply congruence.
1 point
Translates the figure correctly.
1 point
Rotates the figure correctly.
Here is a diagram showing all of the congruent angles that are known, all from the
fact that this figure involves two sets of parallel lines.
S
R
∠UTS ≅ ∠SRU
∠RSU ≅ ∠TUS
∠RUS ≅ ∠TSU
T
Along with the fact that the two triangles share
SU, you can determine that the triangles are
congruent by ASA.
U
1 point
Uses parallel line properties to show congruent angles.
1 point
Identifies SU as a common side.
1 point
Determines ASA for congruence.
Copyright © 2014 Pearson Education, Inc. 3
High School: Congruence and Rigid Motion
CONGRUENCE
AND RIGID MOTION
UNIT ASSESSMENT ANSWERS
Total: 28 points
1.
G-CO-2
1 point
A Translation
2.
G-CO-2
1 point
C Reflection
3.
G-CO-2
1 point
B Rotation around the origin by 270°
4.
G-CO-2
1 point
C Reflection across the line y = x – 5
5.
G-CO-6
1 point
D AAS
6.
G-CO-6
1 point
B SSS
7
G-CO-6
1 point
A SAS
8
G-CO-6
1 point
C ASA
9
G-CO-7
3 points
Possible answer:
∆JKL and ∆MNO are congruent because there is a series of rigid motions that maps
the first triangle exactly onto the second.
Translate ∆JKL to the left 6 units, which maps point K directly onto point N.
Rotate the figure around the point (–4, –4) by 90°. This maps J onto M, and L onto O.
Since this series of rigid motions exists, the two triangles must be congruent.
1 point
Uses the logic that rigid motions imply congruence.
1 point
Describes the translation correctly.
1 point
Describes the rotation correctly.
Copyright © 2014 Pearson Education, Inc. 4
High School: Congruence and Rigid Motion
CONGRUENCE
AND RIGID MOTION
10. G-CO-8 4 points
UNIT ASSESSMENT ANSWERS
Because lines AB and DE are parallel, you can mark several pairs of congruent angles.
Line AE is a transversal of the two parallel lines, so ∠BAE and ∠DEA are congruent
since they are opposite interior angles.
Line BD is a transversal of the two parallel lines, so ∠ABD and ∠EDB are congruent
since they are opposite interior angles.
Lines AE and DB intersect at point C, so ∠ACB and ∠ECD are congruent since they are
vertical angles.
B
A
C
E
D
This gives all three pairs of congruent angles for the two triangles, but you don’t
know anything about the side lengths. This leads to the conclusion that the triangles
are similar, but you cannot be sure they are congruent. You could demonstrate this
idea by sliding line DE farther away from point C, but keeping it parallel to line AB.
All of the angle congruencies would remain, but the triangles would be clearly
different sizes, as shown here.
B
A
C
E
D
1 point
Uses parallel lines to mark congruent angles.
1 point
Uses vertical angles.
1 point
Distinguishes between similar and congruent.
1 point
Explicitly says the triangles may not be congruent.
Copyright © 2014 Pearson Education, Inc. 5
High School: Congruence and Rigid Motion
CONGRUENCE
AND RIGID MOTION
11. G-CO-8 3 points
UNIT ASSESSMENT ANSWERS
AC ≅ CD since point C is given to be the midpoint of AD.
Since lines AB and ED are parallel, ∠BAC ≅ ∠EDC since they are opposite interior
angles.
∠ACB ≅ ∠DCE since they are vertical angles.
Therefore ΔABC and ΔDEC are congruent by ASA.
12. G-CO-8 3 points
1 point
Proves ∠BAC ≅ ∠EDC by opposite interior angles.
1 point
Proves ∠ACB ≅ ∠DCE by vertical angles.
1 point
Proves ΔABC and ΔDEC are congruent by ASA.
With the information given, you can show that ΔLHI and ΔLKI are congruent.
Since ∠KLI ≅ ∠HLI, and point I is the midpoint of KH, it is certain that KI = 2,000 ft
and KH = 2,000 ft. The triangles also share the common side LI. ∠LIK is a right angle,
so ∠LIH ≅ ∠LIK since they are supplementary angles. Therefore you can use SAS or
AAS to prove that ΔLHI and ΔLKI are congruent.
The other three triangles come close, but are not necessarily congruent.
•
∠BAC = 23° rules out ΔABC.
•
No angles are known in ΔACD.
•It is not given that ∠EFG is a right angle, so ΔEFG could be skewed from the
shapes of the other triangles.
1 point
Shows that ΔLHI and ΔLKI are congruent.
1 point
Rules out ΔABC and ΔACD as congruent.
1 point
Describes how ΔEFG could be skewed.
Copyright © 2014 Pearson Education, Inc. 6
High School: Congruence and Rigid Motion
CONGRUENCE
AND RIGID MOTION
13. G-CO-3
3 points
UNIT ASSESSMENT ANSWERS
Statement
Always
Sometimes
The image of a rotated line segment is
parallel to the orignal line segment.
14a. G-CO-2
X
The image of a translated circle remains
a circle.
X
The image of a rotated angle remains the
same angle measure.
X
1 point
Marks first statement as Sometimes.
1 point
Marks second statement as Always.
1 point
Marks third statement as Always.
1 point
y
8
6
4
2
–10
–8
–6
–4
C
D
B
A
–2
2
–2
–4
–6
–8
1 point
Copyright © 2014 Pearson Education, Inc. Draws figure ABCD correctly.
7
4
6
8
10
x
Never
High School: Congruence and Rigid Motion
CONGRUENCE
AND RIGID MOTION
14b. G-CO-5
1 point
UNIT ASSESSMENT ANSWERS
Figure A'B'C'D' is in orange.
y
8
6
4
2
–10
–8
–6
–4
C
D
–2
2
–2
4
C'
–4
B
6
8
10
D'
B'
–6
A
x
A'
–8
Draws figure A'B'C'D' correctly.
1 point
14c. G-CO-5
1 point
Figure A"B"C"D" is in red.
y
8
6
4
2
–10
–8
–6
–4
C
D
B
A
A"
D"
–2
2
B"
–2
4
–4
Copyright © 2014 Pearson Education, Inc. C'
B'
–6
–8
1 point
C" 6
Draws figure A"B"C"D" correctly.
8
8
10
D'
x
High School: Congruence and Rigid Motion
CONGRUENCE
AND RIGID MOTION
14d. G-CO-5
1 point
UNIT ASSESSMENT ANSWERS
Figure A'''B'''C'''D''' is in purple.
y
8
6
4
A'"
2
D'"
–10
–8
8
B'"
D
–6
6
–4
C'"
C
B
A
A"
D"
–2
2
B"
–2
4
C" 6
C'
–4
Copyright © 2014 Pearson Education, Inc. 10
D'
B'
–6
A'
–8
1 point
8
Draws figure A'''B'''C'''D''' correctly.
9
x