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SOLUTIONS Triangle Congruency (2.3)
Triangle Congruency
Concept Development
This section should be completed and understood thoroughly before attempting the
homework exercises.
1. Review of Triangle Terminology. Write a good definition (in conditional form) of the
following and draw a labeled sketch.
Name
Definition
Sketch
D
Scalene Triangle
If a triangle has no
congruent sides, then it is
scalene.
E
F
Isosceles Triangle
If a triangle has two
congruent sides, then it is
isosceles
Equilateral Triangle
If all sides of a triangle are
congruent, then it is
equilateral
Acute Triangle
Obtuse Triangle
Right Triangle
If all angles in a triangle
have measures less than 90
degrees, then it is an acute
triangle
If a triangle has an angle
with measure greater than
90 degrees, it is an obtuse
triangle
If a triangle has a right
angle, then it is a right
triangle
SOLUTIONS Triangle Congruency (2.3)
2. Which of the following triangles are possible? Draw a sketch of all possible triangles.
a. An obtuse isosceles triangle
b. A right equilateral triangle
c. A scalene acute triangle
d. An isosceles right triangles
SOLUTIONS Triangle Congruency (2.3)
3. Congruent Triangles. Since triangles are simply enclosed figures made up of only
three sides and three angles, then the following definition should follow:
Definition: If each of three sides of a triangle is congruent to the corresponding
sides of another triangle and the corresponding angles are also congruent, then
the triangles are congruent.
Suppose we are given two triangles (such as the ones below):
W
B
C
U
V
A
Specifically, we know ABC  WVU if all corresponding angles and corresponding
sides are congruent.
To prove that all sides and angles are congruent every time we want to prove two
triangles are congruent would be needlessly time consuming. We therefore need to
understand that there are certain MINIMUM conditions which guarantee congruence.
We will study these minimum conditions in their entirety, but we will begin by
looking at two of these scenarios, each stated as a postulate:
Side-Angle-Side (SAS): -- Postulate
If two sides of one triangle are congruent to two sides of a second triangle AND the included
angles are also congruent, then the triangles are congruent. Place tick marks for this scenario.
C
Given:
AB  DH
A   D
CA  DG
D
A
B
G
Therefore:

_______________
 
H
________________
(Be sure that corresponding vertices are correct in your statement)
SOLUTIONS Triangle Congruency (2.3)
Angle-Side-Angle (ASA): -- Postulate
If two angles of one triangle are equal to two angles of another triangle AND the included sides
are also congruent, then the triangles are congruent. Place tick marks for this scenario.
C
Given:
B   T
C  S
CB  ST
Therefore:
R
T
A
S
B

_______________
 
________________
(Be sure that corresponding vertices are correct in your statement)
4. Practice. Use the given information to complete each statement. If triangles cannot
be shown to be congruent from the information
given, write “cannot be determined.”
C is the midpoint of BE & AD.
a.
ABC  
Reason:
b.
KI  IT ; IE bisects TEK
KIE  
Reason:
SOLUTIONS Triangle Congruency (2.3)
5. Extension from Congruent Triangles. Suppose we look at a scenario where we have
proven ABC  WVU , using SAS, where AB  WV , B  V , and A  W .
Make tick markings on the triangles below.
W
B
C
U
V
A
a. What conclusion can we draw concerning all remaining corresponding parts of
these congruent triangles?
They must also be congruent!
b. We can justify this using the acronym:
C
P
What do you think CPCTC stands for?
Corresponding parts of congruent triangles are congruent
C
T
C
SOLUTIONS Triangle Congruency (2.3)
6. Write a valid proof. Note that you will end up proving that two triangles are
congruent.
Q
Given: AQ  BC
1  2
AP  AR
Prove:
P
1) APQ  ARQ
2) AQP  AQR
R
1
B
2
A
C
Plan: use SAS via subtraction property – will need to also use reflexive property.
Statements
1. AQ  BC
2. 1  2
3. AP  AR
4. BAQ and CAQ are right angles
Reasons
1. Given
2. Given
3. Given
4. If two lines are perpendicular, then
they form right angles
5. BAQ  CAQ
5. If two angles are right angles, then they
are congruent.
6. 3  4
6. Subtraction Property (5, 3)
7. AQ  AQ
7. Reflexive Property
8. APQ  ARQ
8. SAS (7, 6, 3)
9. AQP  AQR
9. CPCTC (step 8)
QED
Please notice that in order to use CPCTC in your proof that you needed to have a
statement about congruent triangles listed in your proof BEFORE the CPTCTC
step!
SOLUTIONS Triangle Congruency (2.3)
7. Write a valid proof.
O
Given: OP  MR
3  4
M  R
P is the midpt. of MR
N
Prove: N  Q
Q
4
3
M
P
R
Plan: Prove MPN and RPQ are congruent triangles by ASA via subtraction property
Statements
Reasons
1. OP  MR
2. 3  4
3. M  R
4. P is the midpt. of MR
1.
2.
3.
4.
5. OPM , OPR are right angles
5. If two lines are perpendicular, then they
form right angles
6. OPM  OPR
7. 5  6
8. MP  PR
9.
PMN  PRQ
10. N  Q
Given
Given
Given
Given
6. If two angles are right angles, then they
are congruent
7. Subtraction Property (2, 6)
8. If a point is a midpoint of a segment,
then it is equidistant from the endpoints of
the segment.
9. ASA (7, 8, 3)
10. CPCTC (9)
QED
SOLUTIONS Triangle Congruency (2.3)
Exercises
Try these problems after you have completed the concept development section.
8. Write a valid proof involving a pair of congruent triangles. Remember, to prove
congruent triangles you need THREE pieces of information regarding congruent
parts; SAS or ASA.
O
Given: PQ  QN and MQ  QO
P
Q
N
Prove: _______________________
M
Plan: Prove triangles PQM and NQO are congruent via SAS via opposite angles
congruent theorem.
Statements
1. PQ  QN
2. MQ  QO
3. PN intersects MO at Q
4. 2  3
5.
PQM  NQO
6. P  N
Reasons
1. Given
2. Given
3. AFD
4. If lines intersect, then opposite angles
are congruent.
5. SAS (2,4,1)
6. CPCTC (5)
QED
After proving congruent triangles, name all the pairs of corresponding parts:
SOLUTIONS Triangle Congruency (2.3)
9. Write a valid proof involving a pair of congruent triangles. Remember, to prove
congruent triangles you need THREE pieces of information regarding congruent
parts; SAS or ASA.
C
Given: B is the midpoint of AC
DB  AC
B
A
Prove: ___________________________
D
Plan: Use SAS and the reflexive property
Statements
1. B is the midpoint of AC
2. DB  AC
1. Given
2. Given
3. 4 and 5 are right angles
3. If two lines are perpendicular, then they
form right angles
4. 4  5
Reasons
4. If two angles are right angles, then they
are congruent
5. BD  BD
5. Reflexive Property
6. AB  BC
6. If a point is a midpoint of a segment,
then it is equidistant from its endpoints
7.
ABD  CBD
7. SAS (6,4,5)
After proving congruent triangles, name the remaining pairs of corresponding parts:
SOLUTIONS Triangle Congruency (2.3)
10. Write a valid proof involving a pair of congruent triangles. Remember, to prove
congruent triangles you need THREE pieces of information regarding congruent
parts; SAS or ASA
S
Given: SZ  TX
SY  TY
T
Y
Prove: SX  TZ
X
Z
Plan: show triangles SXY and TZY are congruent with SAS via subtraction property
and opposite angles congruent theorem.
Statements
Reasons
1. SZ  TX
2. SY  TY
1. Given
2. Given
3. YZ  XY
3. Subtraction Property (1,2)
4. XT intersects SZ at Y
5. 2  3
6.
SYX  TYZ
7. SX  TZ
4. AFD
5. If lines intersect, then opposite angles
are congruent.
6. SAS (3, 5, 2)
7. CPCTC (6)
QED
SOLUTIONS Triangle Congruency (2.3)
11. Write a valid proof. You will need to prove congruent triangles TWICE in to
complete the proof.
G
Given: GR  MR
1  2
R
1
2
K
J
Prove: JGK  JMK
M
Plan: Show triangles RKG and RKM are congruent with SAS and reflexive property,
then show triangles GKJ and MJK are congruent using SAS and CPCTC from
previously proved congruent triangles.
Statements
Reasons
1. GR  MR
2. 1  2
1. Given
2 .Given
3. RK  RK
3. Reflexive Property
4. GRK  MRK
4. SAS (1,2,3)
5. 3  4
5. CPCTC (4)
6. GK  KM
7. JK  JK
8.
KMJ  KGJ
9. JGK  JMK
6. CPCTC (4)
7. Reflexive Property
8. SAS (6,5,7)
9. CPCTC (8)
QED
SOLUTIONS Triangle Congruency (2.3)
12. Consider a triangle whose sides measure 2t 1 , t  5 , and 3t  8 meters.
a. Determine a value for t , making the triangle isosceles.
b. Is it possible to find a value for t which will make the triangle equilateral? If so,
determine its value.