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Testing for Congruent Triangles Examples
1. Why is congruency important? In 1913, Henry Ford began producing
automobiles using an assembly line. When products are mass-produced, each
piece must be interchangeable, so they must have the same size and shape. Each
piece is an exact copy of the others, and any piece can be made to coincide with
the others.
2. Student activity – Have students draw a triangle and cut it out. Use it as a pattern
to draw a second triangle and cut that triangle out. If one triangle is placed on top
of the other, the two coincide or match exactly. This means that each part of the
first triangle matches exactly the corresponding part of the second triangle. You
have made a pair of congruent triangles.
3. If ∆ ABC is congruent to ∆ RST ( ∆ ABC ≅ ∆ RST), the
vertex labeled A corresponds to the vertex labeled R,
vertex B corresponds to S, and vertex C corresponds to T.
This correspondence can be described in terms of angles
and sides as follows.
∠ A corresponds to ∠ R
∠ B corresponds to ∠ S
∠ C corresponds to ∠ T
AB corresponds to RS
BC corresponds to ST
AC corresponds to RT
Since the two triangles match exactly, the
corresponding parts are congruent.
4.
Definition of Congruent Triangles
(CPCTC)
Two triangles are congruent if and
only if their corresponding parts are
congruent.
The abbreviation CPCTC means
Corresponding Parts of Congruent Triangles
are Congruent.
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5. Example – A triangular wedge is used to anchor the seat belts of a car.
a. Draw two identical wedges and label the vertices P, R, and S on one part and
K, L, and M on the other so that ∆ PRS ≅ ∆ KLM. Then mark the congruent
parts.
b. What angle in ∆ PRS is congruent to ∠ K in ∆ KLM? Î ∠ P is congruent to
∠K
c. Which side of ∆ KLM is congruent to PS in ∆ PRS? Î KM is congruent to
RS
Point out the importance of the order of the
letters in a statement of congruence. ∆ PRS
≅ ∆ KLM, but ∆ PRS is not congruent ∆ MLK.
6. Example – The Adams family is having their game room renovated. This room
will have two triangular windows.
a. Draw two identical windows and label the vertices ABC on one window
and DEF on the other so that ∆ ABC ≅ ∆ DEF.
b. What angle in ∆ ABC is congruent to ∠ F in ∆ DEF? Î ∠ C
c. Which side of ∆ DEF is congruent to AC in ∆ ABC? DF
7. Congruence of triangles, like congruence of segments and angles, is reflexive,
symmetric, and transitive.
Theorem
Congruence of triangles is reflexive, symmetric, and transitive.
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a. Prove that congruence of triangles is reflexive.
Given: ∆ XYZ
Prove : ∆ XYZ ≅ ∆ XYZ
Statements
∆ XYZ
∠ X ≅ ∠ X, ∠ Y ≅ ∠ Y, ∠ Z ≅ ∠ Z
XY ≅ XY , YZ ≅ YZ , XZ ≅ XZ
∆ XYZ ≅ ∆ XYZ
Reasons
Given
Congruence of ∠ ’s is reflexive.
Congruence of segments is reflexive.
Definition of congruent triangles
b. Prove that congruence of triangles is Symmetric.
Statements
∆ LMN ≅ ∆ OPQ
∠ L ≅ ∠ O, ∠ M ≅ ∠ P, ∠ N ≅ ∠ Q, LM ≅ OP ,
MN ≅ PQ , LN ≅ OQ
∠ O ≅ ∠ L, ∠ P ≅ ∠ M, ∠ Q ≅ ∠ N
OP ≅ LM , PQ ≅ MN , OQ ≅ LN
∆ OPQ ≅ ∆ LMN
Reasons
Given
CPCTC
Congruence of angles is symmetric.
Congruence of segments is symmetric.
Definition of congruent triangles.
c. Prove that congruence of
triangles is transitive.
Given: ∆ ABC ≅ ∆ DEF, ∆ DEF ≅ ∆ GHI
Prove ∆ ABC ≅ ∆ GHI
Paragraph Proof:
We are given that ∆ ABC ≅ ∆ DEF. By the definition of congruent triangles, the
corresponding parts of the triangles are congruent. So, ∠ A ≅ ∠ D, ∠ B ≅ ∠ E, ∠ C ≅ ∠ F,
AB ≅ DE , BC ≅ EF , and AC ≅ DF . It is also given that ∆ DEF ≅ ∆ GHI, so by the
definition of congruent triangles, ∠ D ≅ ∠ G, ∠ E ≅ ∠ H, ∠ F ≅ ∠ I, DE ≅ GH , EF ≅ HI ,
and DF ≅ GI . Since congruence of angles is transitive, ∠ A ≅ ∠ G, ∠ B ≅ ∠ H, ∠ C ≅ ∠ I.
Congruence of segments is transitive, so AB ≅ GH , BC ≅ HI , and AC ≅ GI . Therefore,
∆ ABC ≅ ∆ GHI by the definition of congruent triangles.
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8.
Side-Side-Side (SSS) Postulate
If the sides of one triangle are congruent to
the sides of a second triangle, then the
triangles are congruent.
The SSS postulate can be used to prove triangles congruent.
9.
Given: AB ≅ DB , AC ≅ CD
Prove: ABC
DBC
Statements
Reasons
AB ≅ DB
Given
AC ≅ CD
Given
BC ≅ CB
Common Side (Reflexive Property)
ABC
DBC
SSS
10. Example – Given ∆ ABC with vertices A(0, 5), B(2, 0), and C(0, 0) and ∆ RST
with vertices R(5, 8), S(5, 3), T(3, 3), show that ∆ ACB ≅ ∆ RST.
Use the distance formula to show that the corresponding sides are
congruent.
AC =
(0 − 0) 2 + (5 − 0) 2
AC =
25 = 5
RS =
(5 − 5) 2 + (8 − 3) 2
RS =
25 = 5
AB =
(0 − 2) 2 + (5 − 0) 2
AB =
29
RT =
(5 − 3) 2 + (8 − 3) 2
RT =
29
CB =
( 0 − 2) 2 + ( 0 − 0) 2
CB =
4 =2
ST =
(5 − 3) 2 + (3 − 3) 2
ST = 4 = 2
All the pairs of corresponding sides are congruent, so ∆ ACB ≅ ∆ RST by
SSS.
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11. Example – Given ∆ PQR with vertices P(3, 4), Q(2, 2), and R(7, 2) and ∆ STU
with vertices S(6, -3), T(4, -2), and U(4, -7), show that ∆ PQR ≅ ∆ STU.
Use the distance formula to show that the corresponding sides are congruent.
PQ =
(3 − 2) 2 + (4 − 2) 2
PQ =
5
PR =
(3 − 7) 2 + (4 − 2) 2
PR =
20 = 2 5
QR =
( 2 − 7 ) 2 + ( 2 − 2) 2
QR =
25 = 5
TU =
(4 − 4) 2 + (−2 − (−7)) 2
TU =
25 = 5
ST =
(6 − 4) 2 = (−3 − (−2)) 2
ST =
5
SU =
(6 − 4) 2 + (−3 − (−7)) 2
SU =
20 = 2 5
All the pairs of corresponding sides are congruent, so ∆ PQR ≅ ∆ STU by SSS.
12. Side-Angle-Side (SAS) Postulate
If two sides and the included angle of one
triangle are congruent to two sides and an
included angle of another triangle, then the
triangles are congruent.
13. Example:
Given: WZ ≅ YZ , VZ ≅ ZX
Prove: ∆ VZW ≅ ∆ XZY
Statements
WZ ≅ YZ , VZ ≅ ZX
∠ WZV ≅ ∠ YZX
∆ VZW ≅ ∆ XZY
Testing for Congruent Triangles
Reasons
Given
Vertical ∠ ’s are ≅
SAS
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14. Example:
Given: AC ≅ CD, BC ≅ CE
Prove: ABC
DEC
Statements
AC ≅ CD
Given
BC ≅ CE
Given
(Included Angle) ACB
ABC
15. Example:
Reasons
DCE Vertical Angles
SAS
DEC
Given: ∠ 1 and ∠ 2 are
right angles, ST ≅ TP
Prove: ∠ 3 ≅ ∠ 4
Statements
16.
∠ 1 and ∠ 2 are right
angles, ST ≅ TP
Given
∠1≅ ∠2
Al rt ∠ ’s are ≅
TR ≅ TR
Congruence of segments is
reflexive
∆STR ≅ ∆PTR
SAS
∠3≅ ∠4
CPCTC
Angle-Side-Angle (ASA)
Testing for Congruent Triangles
Reasons
If two angles and the included side of one
triangle are congruent to two angles and the
included side of another triangle, the triangles
are congruent.
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Given: ∠ Q and ∠ S are
right angles, QR ≅ SR
17. Example:
Prove: ∠ P ≅ ∠ T
Statements
18. Example:
Reasons
QR ≅ SR
Given
∠ PRQ ≅ ∠ TRS
Vertical ∠ ’s are ≅
∠ Q and ∠ S are right
angles
Given
∠Q ≅ ∠S
All rt. ∠ ’s are ≅
∆ PQR ≅ ∆ TRS
ASA
∠P ≅ ∠T
CPCTC
Given: BE bisects AD ,
∠A≅ ∠D
Prove: AB ≅ DC
Statements
Reasons
BE bisects AD , ∠ A ≅ ∠ D Given
∠1≅ ∠2
Vertical ∠ ’s are ≅
AE ≅ ED
Def. of bisector
∆AEB ≅ ∆DEC
ASA
AB ≅ DC
CPCTC
Testing for Congruent Triangles
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October 8, 2001
Name:___________________
Date:____________
Class:___________________
Testing for Congruent Triangles Worksheet
1.
Draw triangles TLA and RSB. Mark the corresponding parts for ∆ TLA ≅ ∆ RSB.
2.
Describe how you would tell if two triangles were congruent.
3.
If two triangles are congruent, what conclusions can you make? Give an example
to illustrate your answer.
Complete each congruence statement.
4.
∆ ARM ≅ ∆ _____
5.
∆ SPT ≅ ∆ _____
Write a congruence statement for the congruent triangles in each diagram.
6.
7.
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Explain why the following pairs of triangles are not congruent.
8.
Prove the following:
9.
Given:
AB || RT
AR ⊥ AB
BT ⊥ RT
AB ≅ RT
AR ≅ TB
Prove: ∆ ABR ≅ ∆TRB
10.
Refer to ∆ALM and ∆PRT . Name one
additional pair of corresponding parts that
need to be congruent in order to prove
that ∆ALM ≅ ∆PRT . What postulate would
you use to prove the triangles are congruent?
Testing for Congruent Triangles
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October 8, 2001
Determine whether each pair of triangles are congruent. If they are congruent,
indicate the postulate that can be used to prove their congruence.
11.
12.
13.
14.
Write a two-column proof.
Given:
MO ≅ PQ
NO bisects MP
Prove: ∆MNO ≅ ∆PNO
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Testing for Congruent Triangles Worksheet Key
1. Draw triangles TLA and RSB. Mark the corresponding parts for ∆ TLA
≅ ∆ RSB.
2. Describe how you would tell if two triangles were congruent. ÎSee if the six
pairs of corresponding parts are congruent
3. If two triangles are congruent, what conclusions can you make? Give an
example to illustrate your answer. ÎThe six pairs of corresponding parts
are congruent. For example, if ∆ABC ≅ ∆RTS , then ∠ A ≅ ∠ R, ∠ B
≅ ∠ T, ∠ C ≅ ∠ S, AB ≡ RT , BC ≅ TS , and AC ≅ RS .
Complete each congruence statement.
4. ∆ ARM ≅ ∆ LEG
5. ∆ SPT ≅ ∆ PSK
Write a congruence statement for the congruent triangles in each diagram.
6.
7.
∆OAB ≅ ∆AOD
∆MIT ≅ ∆NIT
Explain why the following pairs of triangles are not congruent.
8. The congruent parts are not corresponding.
Testing for Congruent Triangles
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Prove the following:
9. Given:
AB || RT
AR ⊥ AB
BT ⊥ RT
AB ≅ RT
AR ≅ TB
Prove: ∆ ABR ≅ ∆TRB
Statement
AB || RT , AR ⊥ AB , BT ⊥ RT
Reason
Given
AB ≅ RT , AR ≅ TB
∠ BAR and ∠ RTB are Right angles
∠ BAR ≅ ∠ RTB
∠ ABR ≅ ∠ TRB
∠ ABR
≅ ∠ TRB
BR ≅ RB
∆ ABR ≅ ∆TRB
⊥ lines form four rt. angles
All rt. Angles are ≅
If 2 || lines are cut by a transversal, alt.
Interior angles are ≅
If 2 angles in a ∆ are ≅ to 2 angles in
another ∆ , the third angles are also ≅
Congruence of segments is reflexive
Definition of congruent triangles
10. Refer to ∆ALM and ∆PRT . Name one
additional pair of corresponding parts that
need to be congruent in order to prove
that ∆ALM ≅ ∆PRT . What postulate
would you use to prove the triangles are
congruent?
AM ≅ PR
ASA
Determine whether each pair of triangles are congruent. If they are congruent,
indicate the postulate that can be used to prove their congruence.
11. ASA
12. SSS
13. Not congruent
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14. Write a two-column proof.
Given:
MO ≅ PQ
NO bisects MP
Prove: ∆MNO ≅ ∆PNO
Statement
MO ≅ PQ
Reason
Given
NO bisects MP
MN ≡ PN
NO ≅ NO
∆MNO ≅ ∆PNO
Definition of bisector
Congruence of segments is reflexive
SSS
Testing for Congruent Triangles
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October 8, 2001
Name: ____________
Date:__________
Class:_____________
Testing for Congruent Triangles Checklist
1.
On questions 1 thru 3, did the student answer each question correctly?
a.
b.
2.
b.
1 out of 2 (5 points)
Yes (5 points)
Yes (5 points)
Yes (5 points)
Yes (5 points)
On questions 11 thru 13, did the student determine which pair of triangles are congruent?
a.
b.
9.
Yes (10 points)
On question 10, did the student name the correct postulate to prove triangles congruent?
a.
8.
1 out of 2 (5 points)
On question 10, did the student name an additional pair of corresponding parts that need to be
congruent in order to prove ∆ ALM ≅ ∆ TRB?
a.
7.
b.
On question 9, did the student write a correct proof?
a.
6.
Yes (10 points)
On question 8, did the student explain why the triangles are not congruent?
a.
5.
1 out of 3 (5 points)
On questions 6 and 7, did the student state the correct congruence statement?
a.
4.
c.
On questions 4 and 5, did the student state the correct congruence statement?
a.
3.
Yes (15 points)
2 out of 3 (10 points)
Yes (15 points)
2 out of 3 (10 points)
c.
1 out of 3 (5 points
On questions 11 thru 13, did the student indicate the correct postulate to prove the congruence?
a.
b.
Yes (15 points)
2 out of 3 (10 points)
c.
1 out of 3 (5 points)
10. On question 14, did the student a correct write a two-column proof?
a.
Yes (5 points)
Total Number of Points _________
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October 8, 2001
NOTE: The sole purpose of this checklist is to aide the teacher in identifying students that need remediation.
It is suggested that teacher’s devise their own point range for determining grades. In addition, some
students need remediation in specific areas. The following checklist provides a means for the teacher to
access which areas need addressing.
1.
Does the student need remediation in content (corresponding parts of congruent triangles) for
questions 1 thru 3? Yes__________ No__________
2.
Does the student need remediation in content (completing congruence statements) for questions 4
and 5? Yes_________ No__________
3.
Does the student need remediation in content (writing congruence statements) for questions 6 and
7? Yes__________ No__________
4.
Does the student need remediation in content (explaining why triangles are not congruent) for
question 8? Yes__________ No__________
5.
Does the student need remediation in content (proving a statement) for question 9?
Yes__________
No__________
6.
Does the student need remediation in content (analyzing congruent triangles) for question 10?
Yes__________ No__________
7.
Does the student need remediation in content (using postulates to prove congruence) for questions
11 thru 13? Yes__________ No__________
8.
Does the student need remediation in content (writing two-column proofs for congruence) for
question 14? Yes__________ No__________
A 85 points and above
B 81 points and above
Sample
Range of
Points!
C 72 points and above
D 63 points and above
F 62 points and below
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October 8, 2001