1. No category

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Name
Class
3.3
Date
Corresponding Parts
of Congruent Figures
Are Congruent
Essential Question: What can you conclude about two figures that are congruent?
Explore
Resource
Locker
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
D
C
E
F
© Houghton Mifflin Harcourt Publishing Company
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 → ¯
BC → ¯
AC → ¯
EF
DF
DE
A
D
→
D
B
→
E
C
→
F
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.
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Reflect
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
1.
△ABC to △DEF also map the sides and angles of △ABC to the corresponding sides and
angles of △DEF, which establishes congruence.
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
M
L
you make any conclusions about corresponding
S
parts? Why or why not?
Yes; since quadrilateral JKLM is congruent to quadrilateral PQRS, there is a sequence of
2.
R
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
D
A
3.5 cm
2.6 cm
F
Step 2 Find the unknown length.
B
DE = AB, and AB = 2.6 cm,
so DE = 2.6 cm.

3.7 cm
42°
73°
65°
E
C
m∠B
Step 1 Find the angle that corresponds to ∠B.
Since △ABC ≅ △DEF, ∠B ≅ ∠ E .
© Houghton Mifflin Harcourt Publishing Company
_
Step 1 Find the side that corresponds to DE .
_ _
Since △ABC ≅ △DEF, AB ≅ DE .
Step 2 Find the unknown angle measure.
m∠B = m∠ E , and m∠ E = 65 °, so m∠B = 65 °.
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Reflect
Discussion
_ _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
3.
correspondence between the triangles is not given,
Q
so you cannot assume ¯
JKand ¯
QR are corresponding
R
J
parts.
P
L
Your Turn
△STU ≅ △VWX. Find the given side length or angle measure.
X
S
32 ft
SU ≅ ¯
VX.
Since △STU ≅ △VWX, ¯
18°
W
16 ft
124°
4. SU
SU = VX = 43 ft.
43 ft
T
5. m∠S
Since △STU ≅ △VWX, ∠S ≅ ∠V.
38°
V
U
m∠S = m∠V = 38°.
Applying the Properties of Congruence
Explain 2
Rigid motions preserve
length and angle. 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 m∠J = m∠K and vice versa.
Properties of Congruence
_ _
AB ≅ AB
_ _
_ _
If AB ≅ CD , then CD ≅ AD .
_ _
_ _
_ _
If AB ≅ CD and CD ≅ EF , then AB ≅ EF .
© Houghton Mifflin Harcourt Publishing Company
Reflexive Property of Congruence
Symmetric Property of Congruence
Transitive Property of Congruence
Example 2

△ABC ≅ △DEF. Find the given side length or angle measure.
AB
B
_ _
Since △ABC ≅ △DEF, AB ≅ DE .
Therefore, AB = DE.
Write an equation.
Subtract 3
x from each side. 8
Divide each side by 2.
(3x + 8) in.
A
3x + 8 = 5x
(5y + 11)°
= 2x
(6y + 2)°
25 in.
C
F
D
(5x) in. 83°
E
4=x
So, AB = 3x + 8 =3(4) + 8 = 12 + 8 = 20 in.
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
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m∠D
Since △ABC ≅ △DEF, ∠ A ≅ ∠D. Therefore, m∠ A = m∠ D.
5y + 11 = 6y + 2
Write an equation.
11 = y + 2
y from each side.
Subtract 5
Subtract 2 from each side.
(
)
9 = y
∠D = (6y + 2)° = 6 ⋅ 9 + 2 ° = 56 ° .
So, m
Your Turn
Quadrilateral GHJK ≅ quadrilateral LMNP. Find the given side length or
angle measure.
G
(4x + 3) cm
H
(9y + 17)°
P
18 cm
L
(10y)°
(6x − 13) cm
K
J
(11y - 1)°
N
6.
LM
¯ ≅ LM
¯.
Since GHJK ≅ LMNP, GH
Therefore, GH = LM.
7.
m∠H
Since quadrilateral GHJK ≅ quadrilateral
LMNP, ∠H ≅ ∠M. Therefore, m∠H = m∠M.
4 x + 3 = 6x - 13 → 8 = x
9y + 17 = 11y - 1 → 9 = y
LM = 6x - 13 = 6(8) - 13 = 35 cm
Explain 3
M
m∠H = (9y + 17)° = (9 ⋅ 9 + 17)° = 98°
Example 3 Write each proof.

A
Given: △ABD ≅ △ACD
_
Prove: D is the midpoint of BC.
B
Statements
1. △ABD ≅ △ACD
_ _
2. BD ≅ CD
_
3. D is the midpoint of BC .
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D
C
Reasons
1. Given
2. Corresponding parts of congruent
figures are congruent.
© Houghton Mifflin Harcourt Publishing Company
Using Congruent Corresponding Parts in a Proof
3. Definition of midpoint.
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B
Given: Quadrilateral JKLM ≅ quadrilateral
NPQR; ∠J ≅ ∠K
R
K
J
Prove: ∠J ≅ ∠P
L
N
M
Statements
Q
P
Reasons
1. Quadrilateral JKLM ≅ quadrilateral NPQR
1. Given
2. ∠J ≅ ∠K
2. Given
3. ∠K ≅ ∠P
3. Corresponding parts of congruent
figures are congruent.
4. Transitive Property of Congruence
4. ∠J ≅ ∠P
V
Your Turn
Write each proof.
8.
Given: △SVT ≅ △SWT
_
Prove: ST bisects ∠VSW.
S
T
W
Statements
Reasons
1. △ SVT ≅ △ SWT
1. Given
2. ∠VST ≅ ∠WST
2. Corresponding parts of congruent
figures are congruent.
© Houghton Mifflin Harcourt Publishing Company
3. ¯
ST bisects ∠VSW.
9.
3. Definition of angle bisector.
Given: Quadrilateral
_ _ ABCD ≅ quadrilateral EFGH;
AD ≅ CD
_ _
Prove: AD ≅ GH
B
A
D
Statements
C
F
E
H
G
Reasons
1. Quadrilateral ABCD ≅ quadrilateral EFGH
2. ¯
AD ≅ ¯
CD
1. Given
2. Given
3. ¯
CD ≅ ¯
GH
3. Corresponding parts of congruent
figures are congruent.
4. ¯
AD ≅ ¯
GH
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4. Transitive Property of Congruence
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