5.5 Using Congruent Triangles
5.5 Goal Show corresponding parts of congruent triangles are congruent.
Key Words • corresponding parts p. 233
Using Congruent Triangles If you know that two triangles are congruent, you can use the definition of congruent triangles from Lesson 5.1 to conclude that the corresponding parts are congruent. If you know TABC c TDEF, then you can conclude: Corresponding Sides
Corresponding Angles
&* c DE &** AB
aA c aD
&* c EF &* BC
aB c aE
&* c DF &** AC
aC c aF
C A
B
D
E F
EXAMPLE
1
Use Corresponding Parts
&* and CD &* bisect In the diagram, AB each other at M. Prove that aA c aB.
C
A M D
B
Solution 1 First sketch the diagram and label any ●
congruent segments and congruent angles.
C
A
2 Because a A and aB are corresponding ●
M
angles in TADM and TBCM, show that TADM c TBCM to prove that a A c aB.
D
Statements
Reasons
&* and CD &* bisect 1. AB
1. Given
B
each other at M. &** c MB &** 2. MA
2. Definition of segment bisector
3. aAMD c aBMC
3. Vertical Angles Theorem
Student Help
&** c MC &** 4. MD
4. Definition of segment bisector
LOOK BACK
5. TADM c TBCM
5. SAS Congruence Postulate
6. aA c aB
6. Corresponding parts of congruent
For the definition of congruent figures, see p. 233.
triangles are congruent.
5.5
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265
Diagrams often show overlapping triangles. It is usually helpful to visualize or redraw the triangles so that they do not overlap. Original diagram
Redrawn diagram
B D
B C
A
C D E
2
TABC and TEBD do not overlap.
Visualize Overlapping Triangles
Sketch the overlapping triangles separately. Mark all congruent angles and sides. Then tell what theorem or postulate you can use to show TJGH c TKHG.
Student Help VISUAL STRATEGY
Drawing overlapping triangles separately makes it easier to see the triangles and correctly mark congruent sides and angles, as shown on p. 232.
E
A
TABC and TEBD overlap.
EXAMPLE
B
J
K
G
H
Solution 1 Sketch the triangles separately and mark any given information. ●
Think of TJGH moving to the left and TKHG moving to the right. K
J
Mark aGJH c aHKG and aJHG c aKGH.
G
H
G
H
2 Look at the original diagram for shared sides, shared angles, ●
or any other information you can conclude.
&** and HG &** are the same side, In the original diagram, GH &** &** so GH c HG . J
K Add congruence marks to GH &* in each triangle.
G
H
G
H
3 You can use the AAS Congruence Theorem to show that ●
TJGH c TKHG.
266
Chapter 5
Congruent Triangles
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EXAMPLE
3
Use Overlapping Triangles
&* c DE &*. Write a proof that shows AB 䊳 Given a ABC c aDEC &* c CE &* CB
Prove
A
D E
B
䊳 AB &* c DE &*
C
Solution 1 Sketch the triangles separately. ●
D
A
Then label the given information and any other information you can conclude from the diagram.
B
In the original diagram, aC is the same in both triangles (aBCA c aECD).
E
C
C
Mark aC c aC.
2 Show TABC c TDEC to prove ●
&* c DE &*. that AB
Statements
Reasons
1. a ABC c aDEC
1. Given
&* c CE &* 2. CB
2. Given
3. aC c aC
3. Reflexive Prop. of Congruence
4. TABC c TDEC
4. ASA Congruence Postulate
&* c DE &* 5. AB
5. Corresponding parts of congruent
triangles are congruent.
Use Overlapping Triangles 1. Tell which triangle congruence
A
B
D
C
theorem or postulate you would &* c CD &*. use to show that AB
Redraw the triangles separately and label all congruences. Explain how to show that the triangles or corresponding parts are congruent. & c KL &* and 2. Given KJ
&* c ML &**. aJ c aL, show NJ
3. Given aSPR c aQRP and
aQ c aS, show TPQR c TRSP. P
K
M J
S
N P
L
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267
5.5 Exercises Guided Practice Vocabulary Check
1. In the diagram, T ABC c T DEF.
Why can you conclude that aCBA c aFED?
Skill Check
2.
C
F
A
B E
D
Visualize It! Tell which diagram correctly represents all the congruences in the original figure. Original figure
D
E
G
F
A.
B.
D
E
D
E
F E
G
F E
F
G
F
Explain how to show that the statement is true. &* c DC &** 4. AB
3. aSTU c aUVS
S
T
5. aJ c aM
A
J
D
N L
V
U
B
M
C
K
Practice and Applications 6. Showing Congruence In the
Extra Practice
P
R
diagram, T JKL c T PQR. Why can you conclude that aK c aQ?
See p. 684.
J
P K
L
Finding Congruent Parts Tell which triangles you need to show are congruent in order to show that the statement is true.
Homework Help
7. aA c aD
C
C
F
L A
Chapter 5
K
J
Example 1: Exs. 15, 16, 19 Example 2: Exs. 10–13 Example 3: Exs. 14–20 268
&* c BA &* 9. DE
8. aJ c aN
Congruent Triangles
B
D
M
N
A
D
B
E
Visualize It! Sketch the overlapping triangles separately. Mark all congruent angles and sides. Then tell what theorem or postulate you can use to show that the triangles are congruent. &* c DA &*, aADB c aCBD 10. BC
&* c HJ &* 11. aE c aH, EF
D
B
E
J G
C
A
Crafts
F
H
String Designs What theorem or postulate can you use to show that the triangles in the string design are congruent? Explain your reasoning. 12. T ABC c TDEF
13. TGHJ c T KHM
C
H
F
J A
E
B
D
M
K
G
STRING DESIGNS The
shape and size of a string design is determined by how many points along a circle are used to create the design.
14. Logical Reasoning Fill in the missing statements and reasons.
Given
䊳 AB &* c AE &*
E
C
aACB c aADE
F
Prove 䊳 aB c aE A
D
B
Statements
Reasons
&* c AE &* 1. AB
1. _________?_________
2. _________?_________
2. Given
3. _________?_________
3. Reflexive Prop. of Congruence
4. T ABC c T AED
4. _________?_________
5. aB c aE
5. _________?_________
Finding Congruent Parts Use the information in the diagram to prove that the statement is true. &* c NM && 16. JK
15. aA c aC
A
B
C
&* c SR &* 17. QT R
J
T
M
L K D
N
5.5
P
U
Using Congruent Triangles
S
269
18. Argyle Patterns In the argyle pattern shown below,
IStudent Help
&** c VW &** c XY &* c YZ &* and aUVW c aXYZ. Prove that UV && c ZX WU &*.
ICLASSZONE.COM
HOMEWORK HELP Extra help with problem solving in Ex. 18 is at classzone.com
W
Y
U
Z
V
X
Logical Reasoning In Exercises 19 and 20, fill in the missing statements and reasons. 19.
&* and AE &* bisect each other at C. Given 䊳 BD Prove 䊳 aA c aE
E C
B A
Statements
Reasons
&* and AE &* bisect each 1. BD
1. _________?_________
other at C.
Student Help
20.
&* c DC &* 2. BC
2. _________?_________
3. _________?_________
3. Def. of segment bisector
4. aBCA c aDCE
4. _________?_________
5. T ABC c TEDC
5. _________?_________
6. aA c aE
6. _________?_________
Given
䊳 JK &* ∏ LK &*, ML & & ∏ KL &*
K
L
J
M
& c MK && JL
STUDY TIP
If you get stuck, remember that you know the given information and what you are trying to prove. You can fill those in first, then go back to the other steps.
&* c ML & & Prove 䊳 JK
Statements
Reasons
1. _________?_________
1. Given
&* ∏ LK &*, ML & & ∏ KL &* 2. JK
2. _________?_________
3. _________?_________
3. ∏ lines form right angles.
4. TJKL and TMLK are
4. _________?_________
right triangles.
270
Chapter 5
5. _________?_________
5. Reflexive Prop. of Congruence
6. TJKL c TMLK
6. _________?_________
7. _________?_________
7. _________?_________
Congruent Triangles
D
&* 储 PN &*, 21. Challenge In the figure at the right, LK &* c ML &* c LP &*. What aLJK c aPMN, and JM theorem or postulate can be used to show that TJKL c TMNP ? Explain.
Standardized Test Practice
N
K
J
M
L
P
&* c CB &* and 22. Multiple Choice In the diagram, suppose that AD
aBCA c aDAC. Which triangles can you use to prove that aEBA c aEDC ?
A
T ABC and TCDA
B
T ABE and TCDE
C D
T DEB and TAEC
A
B E D C
Not enough information
&* c CE &* and 23. Multiple Choice In the diagram, suppose that AE &* c DE &*. Which triangles can you use to prove that AB &* c CD &** ? BE
Mixed Review
F
T ABC and TCDA
G H J
T ABE and TCDE
A B E
T DEB and TAEC D
Not enough information
C
&( bisects aABC. Find the Angle Bisectors In the diagram, BD value of x. (Lesson 2.2) 24.
25.
A
A
26.
A
D 25ⴗ
B (x ⴚ 3)ⴗ
3x ⴗ
D
54ⴗ
C
C
B
B
(5x ⴙ 5)ⴗ D (6x ⴚ 1)ⴗ C
Perpendicular Lines Find the value of x, given that p ∏ q. (Lesson 3.2) 27.
48ⴗ xⴗ
Algebra Skills
28.
p q
p
29.
p
(8x ⴙ 2)ⴗ 37ⴗ (4x ⴙ 1)ⴗ
q q
Solving Equations Solve the equation. (Skills Review, p. 673) 30. x ⫹ 5 ⫽ 8
31. 7x ⫽ ⫺63
32. 4x ⫺ 9 ⫽ 23
33. 11 ⫹ 3x ⫽ 32
34. 5x ⫺ 3x ⫹ 10 ⫽ 24
35. x ⫹ 2x ⫺ 8 ⫽ 19
5.5
Using Congruent Triangles
271
Key Words • corresponding parts p. 233
Using Congruent Triangles If you know that two triangles are congruent, you can use the definition of congruent triangles from Lesson 5.1 to conclude that the corresponding parts are congruent. If you know TABC c TDEF, then you can conclude: Corresponding Sides
Corresponding Angles
&* c DE &** AB
aA c aD
&* c EF &* BC
aB c aE
&* c DF &** AC
aC c aF
C A
B
D
E F
EXAMPLE
1
Use Corresponding Parts
&* and CD &* bisect In the diagram, AB each other at M. Prove that aA c aB.
C
A M D
B
Solution 1 First sketch the diagram and label any ●
congruent segments and congruent angles.
C
A
2 Because a A and aB are corresponding ●
M
angles in TADM and TBCM, show that TADM c TBCM to prove that a A c aB.
D
Statements
Reasons
&* and CD &* bisect 1. AB
1. Given
B
each other at M. &** c MB &** 2. MA
2. Definition of segment bisector
3. aAMD c aBMC
3. Vertical Angles Theorem
Student Help
&** c MC &** 4. MD
4. Definition of segment bisector
LOOK BACK
5. TADM c TBCM
5. SAS Congruence Postulate
6. aA c aB
6. Corresponding parts of congruent
For the definition of congruent figures, see p. 233.
triangles are congruent.
5.5
Using Congruent Triangles
265
Diagrams often show overlapping triangles. It is usually helpful to visualize or redraw the triangles so that they do not overlap. Original diagram
Redrawn diagram
B D
B C
A
C D E
2
TABC and TEBD do not overlap.
Visualize Overlapping Triangles
Sketch the overlapping triangles separately. Mark all congruent angles and sides. Then tell what theorem or postulate you can use to show TJGH c TKHG.
Student Help VISUAL STRATEGY
Drawing overlapping triangles separately makes it easier to see the triangles and correctly mark congruent sides and angles, as shown on p. 232.
E
A
TABC and TEBD overlap.
EXAMPLE
B
J
K
G
H
Solution 1 Sketch the triangles separately and mark any given information. ●
Think of TJGH moving to the left and TKHG moving to the right. K
J
Mark aGJH c aHKG and aJHG c aKGH.
G
H
G
H
2 Look at the original diagram for shared sides, shared angles, ●
or any other information you can conclude.
&** and HG &** are the same side, In the original diagram, GH &** &** so GH c HG . J
K Add congruence marks to GH &* in each triangle.
G
H
G
H
3 You can use the AAS Congruence Theorem to show that ●
TJGH c TKHG.
266
Chapter 5
Congruent Triangles
IStudent Help ICLASSZONE.COM
MORE EXAMPLES More examples at classzone.com
EXAMPLE
3
Use Overlapping Triangles
&* c DE &*. Write a proof that shows AB 䊳 Given a ABC c aDEC &* c CE &* CB
Prove
A
D E
B
䊳 AB &* c DE &*
C
Solution 1 Sketch the triangles separately. ●
D
A
Then label the given information and any other information you can conclude from the diagram.
B
In the original diagram, aC is the same in both triangles (aBCA c aECD).
E
C
C
Mark aC c aC.
2 Show TABC c TDEC to prove ●
&* c DE &*. that AB
Statements
Reasons
1. a ABC c aDEC
1. Given
&* c CE &* 2. CB
2. Given
3. aC c aC
3. Reflexive Prop. of Congruence
4. TABC c TDEC
4. ASA Congruence Postulate
&* c DE &* 5. AB
5. Corresponding parts of congruent
triangles are congruent.
Use Overlapping Triangles 1. Tell which triangle congruence
A
B
D
C
theorem or postulate you would &* c CD &*. use to show that AB
Redraw the triangles separately and label all congruences. Explain how to show that the triangles or corresponding parts are congruent. & c KL &* and 2. Given KJ
&* c ML &**. aJ c aL, show NJ
3. Given aSPR c aQRP and
aQ c aS, show TPQR c TRSP. P
K
M J
S
N P
L
5.5
R
Using Congruent Triangles
267
5.5 Exercises Guided Practice Vocabulary Check
1. In the diagram, T ABC c T DEF.
Why can you conclude that aCBA c aFED?
Skill Check
2.
C
F
A
B E
D
Visualize It! Tell which diagram correctly represents all the congruences in the original figure. Original figure
D
E
G
F
A.
B.
D
E
D
E
F E
G
F E
F
G
F
Explain how to show that the statement is true. &* c DC &** 4. AB
3. aSTU c aUVS
S
T
5. aJ c aM
A
J
D
N L
V
U
B
M
C
K
Practice and Applications 6. Showing Congruence In the
Extra Practice
P
R
diagram, T JKL c T PQR. Why can you conclude that aK c aQ?
See p. 684.
J
P K
L
Finding Congruent Parts Tell which triangles you need to show are congruent in order to show that the statement is true.
Homework Help
7. aA c aD
C
C
F
L A
Chapter 5
K
J
Example 1: Exs. 15, 16, 19 Example 2: Exs. 10–13 Example 3: Exs. 14–20 268
&* c BA &* 9. DE
8. aJ c aN
Congruent Triangles
B
D
M
N
A
D
B
E
Visualize It! Sketch the overlapping triangles separately. Mark all congruent angles and sides. Then tell what theorem or postulate you can use to show that the triangles are congruent. &* c DA &*, aADB c aCBD 10. BC
&* c HJ &* 11. aE c aH, EF
D
B
E
J G
C
A
Crafts
F
H
String Designs What theorem or postulate can you use to show that the triangles in the string design are congruent? Explain your reasoning. 12. T ABC c TDEF
13. TGHJ c T KHM
C
H
F
J A
E
B
D
M
K
G
STRING DESIGNS The
shape and size of a string design is determined by how many points along a circle are used to create the design.
14. Logical Reasoning Fill in the missing statements and reasons.
Given
䊳 AB &* c AE &*
E
C
aACB c aADE
F
Prove 䊳 aB c aE A
D
B
Statements
Reasons
&* c AE &* 1. AB
1. _________?_________
2. _________?_________
2. Given
3. _________?_________
3. Reflexive Prop. of Congruence
4. T ABC c T AED
4. _________?_________
5. aB c aE
5. _________?_________
Finding Congruent Parts Use the information in the diagram to prove that the statement is true. &* c NM && 16. JK
15. aA c aC
A
B
C
&* c SR &* 17. QT R
J
T
M
L K D
N
5.5
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Using Congruent Triangles
S
269
18. Argyle Patterns In the argyle pattern shown below,
IStudent Help
&** c VW &** c XY &* c YZ &* and aUVW c aXYZ. Prove that UV && c ZX WU &*.
ICLASSZONE.COM
HOMEWORK HELP Extra help with problem solving in Ex. 18 is at classzone.com
W
Y
U
Z
V
X
Logical Reasoning In Exercises 19 and 20, fill in the missing statements and reasons. 19.
&* and AE &* bisect each other at C. Given 䊳 BD Prove 䊳 aA c aE
E C
B A
Statements
Reasons
&* and AE &* bisect each 1. BD
1. _________?_________
other at C.
Student Help
20.
&* c DC &* 2. BC
2. _________?_________
3. _________?_________
3. Def. of segment bisector
4. aBCA c aDCE
4. _________?_________
5. T ABC c TEDC
5. _________?_________
6. aA c aE
6. _________?_________
Given
䊳 JK &* ∏ LK &*, ML & & ∏ KL &*
K
L
J
M
& c MK && JL
STUDY TIP
If you get stuck, remember that you know the given information and what you are trying to prove. You can fill those in first, then go back to the other steps.
&* c ML & & Prove 䊳 JK
Statements
Reasons
1. _________?_________
1. Given
&* ∏ LK &*, ML & & ∏ KL &* 2. JK
2. _________?_________
3. _________?_________
3. ∏ lines form right angles.
4. TJKL and TMLK are
4. _________?_________
right triangles.
270
Chapter 5
5. _________?_________
5. Reflexive Prop. of Congruence
6. TJKL c TMLK
6. _________?_________
7. _________?_________
7. _________?_________
Congruent Triangles
D
&* 储 PN &*, 21. Challenge In the figure at the right, LK &* c ML &* c LP &*. What aLJK c aPMN, and JM theorem or postulate can be used to show that TJKL c TMNP ? Explain.
Standardized Test Practice
N
K
J
M
L
P
&* c CB &* and 22. Multiple Choice In the diagram, suppose that AD
aBCA c aDAC. Which triangles can you use to prove that aEBA c aEDC ?
A
T ABC and TCDA
B
T ABE and TCDE
C D
T DEB and TAEC
A
B E D C
Not enough information
&* c CE &* and 23. Multiple Choice In the diagram, suppose that AE &* c DE &*. Which triangles can you use to prove that AB &* c CD &** ? BE
Mixed Review
F
T ABC and TCDA
G H J
T ABE and TCDE
A B E
T DEB and TAEC D
Not enough information
C
&( bisects aABC. Find the Angle Bisectors In the diagram, BD value of x. (Lesson 2.2) 24.
25.
A
A
26.
A
D 25ⴗ
B (x ⴚ 3)ⴗ
3x ⴗ
D
54ⴗ
C
C
B
B
(5x ⴙ 5)ⴗ D (6x ⴚ 1)ⴗ C
Perpendicular Lines Find the value of x, given that p ∏ q. (Lesson 3.2) 27.
48ⴗ xⴗ
Algebra Skills
28.
p q
p
29.
p
(8x ⴙ 2)ⴗ 37ⴗ (4x ⴙ 1)ⴗ
q q
Solving Equations Solve the equation. (Skills Review, p. 673) 30. x ⫹ 5 ⫽ 8
31. 7x ⫽ ⫺63
32. 4x ⫺ 9 ⫽ 23
33. 11 ⫹ 3x ⫽ 32
34. 5x ⫺ 3x ⫹ 10 ⫽ 24
35. x ⫹ 2x ⫺ 8 ⫽ 19
5.5
Using Congruent Triangles
271
