CorrectionKey=NL-A;CA-A 3 . 3 DO NOT EDIT- …
[Pages:12]3.3 L E S S O N
Corresponding Parts of Congruent Figures Are Congruent
Name
Class
Date
3.3 Corresponding Parts of Congruent Figures Are Congruent
Essential Question: What can you conclude about two figures that are congruent?
Resource Locker
Common Core Math Standards
The student is expected to:
COMMON CORE
G-CO.B.7
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
Mathematical Practices
COMMON CORE
MP.2 Reasoning
Language Objective
Have students fill in sentence stems to explain why figures are congruent or noncongruent.
ENGAGE
Essential Question: What can you conclude about two figures that are congruent?
The corresponding parts are congruent, and relationships within the figures, such as relative distances between vertices, are equal.
Explore Exploring Congruence of Parts of Transformed Figures
You will investigate some conclusions you can make when you know that two figures are congruent.
A Fold a sheet of paper in half. Use a straightedge to draw a triangle on the folded sheet.
Then cut out the triangle, cutting through both layers of paper to produce two congruent triangles. Label them ABC and DEF, as shown.
A B
C
DE
F
B Place the triangles next to each other on a desktop. Since the triangles are congruent, there
must be a sequence of rigid motions that maps ABC to DEF. Describe the sequence of rigid motions.
A translation (perhaps followed by a rotation) maps ABC to DEF.
C The same sequence of rigid motions that maps ABC to DEF maps parts of ABC to
parts of DEF. Complete the following.
_ AB
? DE
_ BC
? EF
_ AC
? DF
A D
B E
C F
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PREVIEW: LESSON PERFORMANCE TASK
View the online Engage. Discuss the photo and ask students to identify congruent shapes in the design. Then preview the Lesson Performance Task.
D What does Step C tell you about the corresponding parts of the two triangles? Why?
The corresponding parts are congruent because there is a sequence of rigid motions that maps each side or angle of ABC to the corresponding side or angle of DEF.
Module 3
GE_MNLESE385795_U1M03L3.indd 139
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139
Date Class
3.3 Corresponding Parts Name of Congruent Figures ReLsoocukrecre
Are Congruent Exploring Congruence of Parts of Transformed EsCOsCMeOMRnOENtialGcQ-oCnuOg.eBrs.u7teiUonstnei:ftWahnehddaoetnfcilnyaintiifoycnoorourfeccsoopnongncrdluuinedngecpeaabiinrostuoetrfmstiwds eoosffariigngudidrcemosrortethisoapntosnatdroeinschgoopnwagitrrhsuaoetfntawtn?ogltersiaanrgeles are congruent. Figures You wEixlFTltpoirhnlidlaevonneasgrctsleiuehgstae.teoeLtuasotboftmehpleeatphtcreeoirmnancinlgulhseAia,olBcnfu.CstUytaiosnneugdactahsntrrDomauEigagFkhh,etaebwsdohgstheehnotlowayyondeu.rraskwnoofawptartiphaeanrtgttlwoe n. t AB
DE C
F
Lesson 3
HARDCOVER PAGES 123132
04/04/14
Turn to these pages to find this lesson in the hardcover student
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that Lesson
3
edition.
139 Module 3
01/04/14 10:58 PM
7:00 PM
GE_MNLESE385795_U1M03L3 139
139 Lesson 3.3
Reflect
1. If you know that ABC DEF, what six congruence statements about segments and angles can you write? Why? ? AB ? DE, ? BC ? EF, ? AC ? DF, A D, B E, C F. The rigid motions that map
ABC to DEF also map the sides and angles of ABC to the corresponding sides and
angles of DEF, which establishes congruence.
2. Do your findings in this Explore apply to figures
J
K
other than triangles? For instance, if you know that
P
Q
quadrilaterals JKLM and PQRS are congruent, can
you make any conclusions about corresponding
M
L
parts? Why or why not?
S
R
Yes; since quadrilateral JKLM is congruent to quadrilateral PQRS, there is a sequence of
rigid motions that maps JKLM to PQRS. This same sequence of rigid motions maps sides
and angles of JKLM to the corresponding sides and angles of PQRS.
Explain 1 Corresponding Parts of Congruent Figures Are Congruent
The following true statement summarizes what you discovered in the Explore.
Corresponding Parts of Congruent Figures Are Congruent If two figures are congruent, then corresponding sides are congruent and corresponding angles are congruent.
Example 1 ABC DEF. Find the given side length or angle measure.
DE
_ Step 1 Find the side that corresponds to DE.
_ _ Since ABC DEF, AB DE.
Step 2 Find the unknown length.
DE = AB, and AB = 2.6 cm, so DE = 2.6 cm.
A 2.6 cm
B
3.7 cm
D 3.5 cm 73?
F 42?
65? E
C
mB
Step 1 Find the angle that corresponds to B. Since ABC DEF, B E .
Step 2 Find the unknown angle measure. mB = m E , and m E = 65 ?, so mB = 65 ?.
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EXPLORE
Exploring Congruence of Parts of Transformed Figures
QUESTIONING STRATEGIES
When you are given two congruent triangles, how many pairs of corresponding parts--angles and sides--are there? 6; 3 angles and 3 sides
EXPLAIN 1
Corresponding Parts of Congruent Figures Are Congruent
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Have a student read the statement about
Corresponding Parts of Congruent Figures. Discuss the meaning of the statement for general figures and then in terms of two triangles. Emphasize that the statement is a biconditional, an if-and-only-if statement that is true when read as an if-then statement in either direction.
Module 3
140
Lesson 3
PROFESSIONAL DEVELOPMENT
GE_MNLESE385795_U1M03L3 140
Math Background
In this lesson, students learn that if two figures (including triangles) are congruent, then corresponding pairs of sides and corresponding pairs of angles of the figures are congruent. This follows readily from the rigid-motion definition of congruence and from the statement that Corresponding Parts of Congruent Figures Are Congruent. This statement is a biconditional, a statement that is true in either direction. That is, if corresponding pairs of sides and corresponding pairs of angles in two figures are congruent, then the figures are congruent.
4/5/14 2:04 PM
Corresponding Parts of Congruent Figures are Congruent 140
QUESTIONING STRATEGIES
How do you determine which sides of two congruent figures correspond? Use the order of letters in the congruence statement. The first letters correspond, the last letters correspond, and the other letters correspond in the same order.
Reflect
3. Disc_ussio_ n The triangles shown in the figure are congruent. Can you conclude that JK QR? Explain.
K
No; the segments appear to be congruent, but the
Q
P
J
L
correspondence between the triangles is not given,
so you cannot assume ? JK and ? QR are corresponding R
parts.
VISUAL CUES
Have each student make a poster illustrating the concept of congruent figures. The illustrations should be labeled to show which pairs of corresponding parts are congruent. Have them show both examples and non-examples of congruent figures in the poster.
Your Turn
STU VWX. Find the given side length or angle measure.
S
16 ft W
124?
T
38?
32 ft 18? 43 ft
X 4. SU Since STU VWX, ? SU ? VX. SU = VX = 43 ft. 5. mS
U
V
Since STU VWX, S V.
mS = mV = 38?.
EXPLAIN 2
Applying the Properties of Congruence
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Suggest that students list all the congruencies
that relate the parts of the figures and mark the figures to show them. Once they have clearly represented the corresponding parts, they can more easily answer the questions.
QUESTIONING STRATEGIES
How could you use transformations to decide whether two figures are congruent? You could use transformations to create all pairs of corresponding parts congruent. Then the statement applies because if corresponding parts of congruent figures are congruent, then the figures are congruent.
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Explain 2 Applying the Properties of Congruence
Rigid motions pr_ eserve_length and angle measure. This means that congruent segments have the same length, so UV XY implies UV = XY and vice versa. In the same way, congruent angles have the same measure, so J K implies mJ = mK and vice versa.
Properties of Congruence Reflexive Property of Congruence Symmetric Property of Congruence Transitive Property of Congruence
_ _ AB AB
_ _
_ _
If AB CD, then CD AD.
__ __
_ _
If AB CD and CD EF, then AB EF.
Example 2 ABC DEF. Find the given side length or angle measure.
AB
_ _ Since ABC DEF, AB DE. Therefore, AB = DE.
B
(6y + 2)?
F
(3x + 8) in.
25 in.
D
A
Write an equation.
3x + 8 = 5x
(5y + 11)?
(5x) in. 83?
C
E
Subtract 3x from each side.
8 = 2x
Divide each side by 2.
4 = x
So, AB = 3x + 8 = 3(4) + 8 = 12 + 8 = 20 in.
Module 3
141
Lesson 3
COLLABORATIVE LEARNING
Small Group Activity GE_MNLESE385795_U1M03L3 141
5/14/14 6:03 PM
Have each student draw a pair of congruent figures on paper. Instruct them to switch papers and to write a congruence statement for the pair of figures. Then have them switch papers several more times within groups, write new congruence statements that fit the pair of figures, and list the congruent pairs of corresponding parts of the figures.
141 Lesson 3.3
mD
Since ABC DEF, A D. Therefore, m A = mD.
Write an equation.
5y + 11 = 6y + 2
Subtract 5y from each side.
11 = y + 2
Subtract 2 from each side.
9= y
( ) So, mD = (6y + 2)? = 6 9 + 2 ? = 56 ?.
Your Turn
Quadrilateral GHJK quadrilateral LMNP. Find the given side length or angle measure.
G (4x + 3) cm H
(9y + 17)?
P
18 cm
(10y)?
L
(6x - 13) cm K
J
(11y - 1)? M
N
6. LM Since GHJK LMNP, G?H L?M. Therefore, GH = LM.
4x + 3 = 6x - 13 8 = x LM = 6x - 13 = 6(8) - 13 = 35 cm
7. mH Since quadrilateral GHJK quadrilateral LMNP, H M. Therefore, mH = mM. 9y + 17 = 11y - 1 9 = y
mH = (9y + 17)? = (9 9 + 17)? = 98?
Explain 3 Using Congruent Corresponding Parts in a Proof
Example 3 Write each proof.
A
Given: ABD ACD
_ Prove: D is the midpoint of BC.
B
D
C
Statements 1. ABD ACD
1. Given
Reasons
_ _ 2. BD CD
_ 3. D is the midpoint of BC.
2. Corresponding parts of congruent figures are congruent.
3. Definition of midpoint.
Module 3
142
Lesson 3
DIFFERENTIATE INSTRUCTION
Technology GE_MNLESE385795_U1M03L3.indd 142
Have students use geometry software to create designs using congruent triangles. They should arrange multiple congruent triangles using different colors, positions, and orientations. Ask them to make three separate designs: one using congruent equilateral triangles, one using congruent isosceles triangles, and one using congruent scalene triangles.
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AVOID COMMON ERRORS
Students may correctly solve for a variable but then incorrectly give the value of the variable as a side length or angle measure. Remind them to examine the diagram carefully; sometimes a side length or angle measure is described by an expression containing a variable, not by the variable alone.
EXPLAIN 3
Using Congruent Corresponding Parts in a Proof
INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Encourage students to use geometry software
to reflect the triangle with the given conditions and then to verify that corresponding congruent parts have equal measure.
CONNECT VOCABULARY
In this lesson, students learn the Corresponding Parts of Congruent Figures Are Congruent. Although acronyms (such as CPCTC) may be helpful to some students when referring to statements, postulates, or theorems, such devices may be a bit more difficult for English Learners at the Emerging level. Consider making a poster or having students create or copy a list of theorems, along with their meanings, for them to refer to in this module. Students may want to come up with a mnemonic for the CPCTC itself, such as Cooks Pick Carrots Too Carefully.
04/04/14 7:02 PM
Corresponding Parts of Congruent Figures are Congruent 142
QUESTIONING STRATEGIES
Why do pairs of corresponding congruent parts have equal measure? Since rigid motions preserve angle measure and length, and since there is a sequence of rigid motions that maps a figure to a congruent figure, pairs of corresponding parts must have equal measure.
ELABORATE
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 When examining congruent figures, students
can see how each vertex is mapped to its corresponding vertex by designating corresponding vertices in the same color and using a different color for each pair of corresponding vertices. Students can also highlight pairs of corresponding sides in the same color, using a different color for each pair.
QUESTIONING STRATEGIES
Can you say two figures are congruent if their corresponding angles have the same measure? Explain. No. You must also determine that the corresponding sides have the same measure.
Can you say that a pair of corresponding sides of two congruent figures has equal measure? Yes. If the figures are congruent, then each pair of corresponding sides is congruent and therefore has equal measure.
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B Given: Quadrilateral JKLM quadrilateral J
NPQR; J K Prove: J P
Statements 1. Quadrilateral JKLM quadrilateral NPQR 2. J K 3. K P 4. J P
K
R
Q
L
N
P
M
1. Given
Reasons
2. Given
3.Corresponding parts of congruent figures are congruent.
4.Transitive Property of Congruence
Your Turn
Write each proof. 8. Given: _SVT SWT
Prove: ST bisects VSW.
V
S
T
Statements 1. SVT SWT 2. VST WST
3. ? ST bisects VSW.
W Reasons 1. Given 2. Corresponding parts of congruent figures are congruent. 3. Definition of angle bisector.
9. Given: Q_uadri_ lateral ABCD quadrilateral EFGH;
A
BE
F
A_D C_D
Prove: AD GH
D
C
H
G
Statements 1. Quadrilateral ABCD quadrilateral EFGH 2. ? AD ? CD 3. ? CD ? GH
4. ? AD ? GH
Reasons 1. Given 2. Given 3. Corresponding parts of congruent
figures are congruent. 4. Transitive Property of Congruence
Module 3
143
Lesson 3
SUMMARIZE THE LESSON
Suppose you know that CBA EFG. What are six congruency statements? C E, B F, A G, C?B E?F, C?A E?G, B?A F?G
LANGUAGE SUPPORT
Connect Vocabulary GE_MNLESE385795_U1M03L3.indd 143
21/03/14 2:11 PM
Have students work in pairs. Provide each student with a protractor and ruler, and ask them to explain why two figures are congruent or noncongruent. Provide students with sentence stems to help them describe the attributes of the figures. For example: "The two (triangles/quadrilaterals/figures) are or are not congruent because their corresponding angles have/don't have equal measures. Angles ___ and ____ are corresponding, and measure _____ degrees. Corresponding sides have equal/not equal lengths." Students work together to complete the sentences.
143 Lesson 3.3
Elaborate
10. A student claims that any two congruent triangles must have the same perimeter. Do you agree? Explain. Yes; since the corresponding sides of congruent triangles are congruent, the sum of the
lengths of the sides (perimeter) must be the same for both triangles.
11. If PQR is a right triangle and PQR XYZ, does XYZ have to be a right triangle? Why or why not? Yes; since PQR is a right triangle, one of its angles is a right angle. Since corresponding
parts of congruent figures are congruent, one of the angles of XYZ must also be a right
angle, which means XYZ is a right triangle.
12. Essential Question Check-In Suppose you know that pentagon ABCDE is congruent to pentagon FGHJK. How many additional congruence statements can you write using corresponding parts of the pentagons? Explain. There are five statements using the congruent corresponding sides and five statements
using the congruent corresponding angles.
Evaluate: Homework and Practice
1. Danielle finds that she can use a translation and a reflection to make
quadrilateral ABCD fit perfectly on top of quadrilateral WXYZ. What
congruence statements can Danielle write using the sides and angles of
the quadrilaterals? Why?
A
B
? Online Homework ? Hints and Help ? Extra Practice
Y
C
Z
D
X
W
The same sequence of rigid motions that maps ABCD to WXYZ also maps sides and angles
of ABCD to corresponding sides and angles of WXYZ. Therefore, those sides and angles are congruent:? AB ? WX, ? BC ? XY,_ CD ? YZ, ? AD ? WZ, A W, B X, C Y, D Z.
DEF GHJ. Find the given side length or angle measure. D
19 ft
112?
E
31 ft
FJ
42 ft 25?
G 43?
H
2. JH Since DEF GHJ, F?E J?H. FE = JH = 31 ft, so JH = 31 ft.
3. mD Since DEF GHJ, D G.
mD = mG = 43?
Module 3
144
Lesson 3
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EVALUATE
ASSIGNMENT GUIDE
Concepts and Skills
Explore Exploring Congruence of Parts of Transformed Figures
Example 1 Corresponding Parts of Congruent Figures are Congruent
Example 2 Applying the Properties of Congruence
Example 3 Using Congruent Corresponding Parts in a Proof
Practice
Exercises 1
Exercises 2?5, 10?13 Exercises 6?9
Exercises 14?16
INTEGRATE MATHEMATICAL PRACTICES
Focus on Math Connections
MP.1 Have students consider whether two
quadrilaterals, both with side lengths of 1 foot on each side, are congruent. Students should recognize that the description is that of a rhombus. Demonstrate that a box with an open top and bottom lying on its side is not rigid, and although the side lengths stay the same when one side is pushed, the angles change. Thus it is possible for the two figures described to have different angle measures and not be congruent.
Exercise GE_MNLESE385795_U1M03L3.indd 144 Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
1 2?5 6?9 14?16 10?13, 17?18 19?22
1 Recall of Information 1 Recall of Information 1 Recall of Information 2 Skills/Concepts 2 Skills/Concepts
2 Skills/Concepts
MP.6 Precision MP.2 Reasoning MP.4 Modeling MP.3 Logic MP.2 Reasoning
MP.4 Modeling
21/03/14 2:11 PM
Corresponding Parts of Congruent Figures are Congruent 144
INTEGRATE MATHEMATICAL PRACTICES
Focus on Communication
MP.3 Have students compare their congruence
statements for a given diagram, and ask them to write other correct congruence statements for the same diagram. Then have them write a congruence statement that is not correct for the diagram and explain why it is not correct.
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KLMN PQRS. Find the given side length or angle measure.
S
K
L
2.1 cm
75?
79? M
N
2.9 cm
4. mR M R. mM = mR = 79?.
R P
Q 5. PS ? KN ? PS. KN = PS =
2.1 cm
ABC TUV. Find the given side length or angle measure.
A
V (5x + 7) cm U
(4y - 18)?
(6x - 1) cm (3y + 2)? C (6x + 2) cm B
(4y)? T
6. BC ? BC U?V. So, BC = UV. 6x + 2 = 5x + 7 x = 5
So, BC = 6x + 2 = 6(5) + 2 = 30 + 2 = 32 cm.
7. mU B U. So, mB = mU. 3y + 2 = 4y - 18 20 = y
So, mU = (4y - 18)? =
(4 20 - 18)? = 62?.
DEFG KLMN. Find the given side length or angle measure.
D (20x + 12)?
G
(2y + 3)in.
N
(4y - 29) in.
E
68?
M
F
(25x - 8)?
L
K (y + 9) in.
8. FG ? FG ? MN. So, FG = MN. 2y + 3 = 4y - 29 16 = y
So, FG = 2y + 3 = 2(16) + 3 = 32 + 3 = 35 in.
9. mD D K. So, mD = mK. 20x + 12 = 25x - 8 4 = x
So, mD = (20x + 12)? = (20 4 + 12)? = 92?.
GHJ PQR and PQR STU. Complete the following using a side or angle of STU. Justify your answers.
10.
_ GH
S?T
11. J U
GHJ STU by the Transitive Prop. of
GHJ STU by the Transitive Prop. of
Cong., and corr. parts of fig. .
Cong., and corr. parts of fig. .
12. GJ = SU GHJ STU by the Transitive Prop.
13. mG = mS GHJ STU by the Transitive Prop. of
of Cong., and corr. parts of fig. .
Cong., and corr. parts of fig. . Cong.
Congruent segments have the same length.
Module 3
145
angles have the same measure.
Lesson 3
Exercise GE_MNLESE385795_U1M03L3.indd 145 Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
23 24?25
26 27
2 Skills/Concepts 3 Strategic Thinking 3 Strategic Thinking 3 Strategic Thinking
MP.2 Reasoning MP.3 Logic MP.6 Precision MP.3 Logic
6/9/15 12:23 AM
145 Lesson 3.3
Write each proof.
14. Given: Q_uadrilatera_l PQTU quadrilateral QRST
P
Q
R
Prove: QT bisects PR.
U
T
S
Statements 1. Quadrilateral PQTU quadrilateral QRST 2. ? PQ ? QR 3. Q is the midpoint of ? PR. 4. ? QT bisects ? PR.
Reasons 1. Given 2. Corr. parts of fig. are 3. Definition of midpoint 4. Definition of segment bisector
15. Given: ABC ADC
_
_
Prove: AC bisects BAD and AC bisects BCD.
A
B C
Statements 1. ABC DEF 2. BAC DAC 3. BCA DCA 4. ? AC bisects BAD and ? AC bisects BCD.
D Reasons
1. Given 2. Corr. parts of fig. are 3. Corr. parts of fig. are 4. Definition of angle bisector
16. Given: Pentagon ABCDE pentagon FGHJK; D E Prove: D K
B
G
A
CH
F
E
DJ
K
Statements 1. Pentagon ABCDE pentagon FGHJK 2. D E 3. E K 4. D K
Reasons 1. Given 2. Given 3. Corr. parts of fig. are 4. Transitive Property of Congruence
Module 3
146
Lesson 3
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AVOID COMMON ERRORS
Students may find the value of a variable or the value of an algebraic expression as the solution to a problem when they are in fact only part of the way through the solving process. Remind students to always go back to the initial question to make sure the answer is the solution to the problem.
GE_MNLESE385795_U1M03L3 146
5/14/14 6:59 PM
Corresponding Parts of Congruent Figures are Congruent 146
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