Download Geometry Triangle Congruence Criteria Unit CO.8 OBJECTIVE #: G

Geometry
OBJECTIVE #:
Triangle Congruence Criteria
Unit CO.8
G.CO.8
OBJECTIVE
 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of
congruence in terms of rigid motion.
BIG IDEA (Why is this included in the curriculum?)
 To determine the minimum criteria necessary to establish triangle congruency.
PREVIOUS KNOWLEDGE (What skills do they need to have to succeed?)
 The student must identify the parts, sides and angles, of a given triangle.
 The student must understand congruent segments have the same length.
 The student must understand congruent angles have the same measure.
 The student must to understand CPCTC.
VOCABULARY USED IN THIS OBJECTIVE (What terms will be essential to understand?)
PREVIOUS VOCABULARY (Terms used but defined earlier)
 Angle: A geometric figure formed by rotating a ray about its initial point.
 Corresponding Parts of Congruent Figures: The sides or angles that are in corresponding
positions when two figures are congruent.
 CPCTC: Corresponding Parts of Congruent Triangles are Congruent.
 Side of a Polygon: The segment formed by connecting two adjacent vertices of a polygon.
 Triangle: A figure formed by three segments joining three non-collinear points, called vertices.
NEW VOCABULARY (New Terms and definitions introduced in this objective)
 AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are congruent to
two angles and the corresponding non-included side of a second triangle, then the two triangles
are congruent.
Notation: ABC  DEF

ASA (Angle-Side-Angle): If two angles and the included side of one triangle are congruent to
two angles and the included side of a second triangle, then the two triangles are congruent.
Notation: ABC  DEF
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Geometry

Triangle Congruence Criteria
Unit CO.8
HL (Hypotenuse-Leg): If the hypotenuse and a leg of a right triangle are congruent to
the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.
Notation: ABC  DEF

SAS (Side-Angle-Side): If two sides and the included angle of one triangle are congruent to two
sides and he included angle of a second triangle, then the two triangles are congruent.
Notation: ABC  DEF

SSS (Side-Side-Side): If three sides of on triangle are congruent to three sides of a second
triangle, then the two triangles are congruent.
Notation: ABC  DEF
SKILLS (What will they be able to do after this objective?)
 Students will be able to identify corresponding angles and sides based on congruence statements.
 Students will be able to develop and write congruence statements for two figures by identifying
corresponding parts.
 Students will be able to determine if two triangles are congruent based on their corresponding parts.
 Students will be able to apply the criteria of AAS, ASA, SAS, SSS, and HL to prove triangle
congruency.
SHORT NOTES (A short summary of notes so that a teacher can get the basics of what is expected.)
 Students should be able to derive the triangle congruence postulates.
 In a right triangle, HL (Hypotenuse – Leg) is a congruence postulate.
 As an additional resource, you may reference Patty Paper Geometry by Michael Serra pg. 125 – 139
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Geometry
Triangle Congruence Criteria
Unit CO.8
ACTIVITY
 The following activity allows students to explore what criteria are necessary for triangle congruence.
Components of Triangles
1
2
3
4
5
a
b
c
e
d
Testing Criteria for Triangle Congruence
1. Given one side (S)
2. Given one angle (A)
3. Given two sides (SS)
4. Given two angles (AA)
 Grab a piece of patty paper and in a corner write the criteria that you are currently using to create the
triangle. In this case put (AA) and then put your name under that.
 Now create a triangle using 1 and 2. Copy 1 onto the piece of patty paper and then, using the
provided sheet, close the triangle using 2. Trace the completed triangle. Label 1 and 2 in the
completed triangle.
 Compare your triangle with the person beside you. Compare it with 4 people around you…
 Repeat these steps but with 3 and 5…
 Discuss with your partner: Do you think AA is a congruence criteria? Why or why not?
5. Given an Angle, the included Side, and then the other Angle. (ASA)
 Grab a piece of patty paper and in a corner write the criteria that you are currently using to create the
triangle. So in this case put (ASA), and then put your name under that.
 Now create a triangle using 3, side d, and 1. The order of these matter… side d must be the included
side (the side directly between 3 and 1). Trace the completed triangle. Label 3 AND side d and
1 in the completed triangle.
 Compare your answer with the person beside you. Compare it with 4 people around you…
 Repeat these steps but with 4, side b, and 2…
 Discuss with your partner: Do you think ASA is a congruence criteria? Why or why not?
6. Given a Side, the included Angle, and then the other Side. (SAS)
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Unit CO.8
 Grab a piece of patty paper and in a corner write the criteria that you are currently using to
create the triangle. So in this case put (SAS), and then put your name under that.
 Now create a triangle using side c, 3, and side d. The order of these matter… 3 must be the included
angle (the angle directly between side c and side d). Trace the completed triangle. Label side c, 3,
side d in the completed triangle.
 Compare your answer with the person beside you. Compare it with 4 people around you…
 Repeat these steps but with side b, 2, and side c.
 Discuss with your partner: Do you think SAS is a congruence criteria? Why or why not?
7. Given a Side, the included Angle, and then the other Side. (SAA) (the same as AAS just reverse order)
 Grab a piece of patty paper; in a corner write the criteria that you are currently using to create the
triangle. So in this case put (SAA), and then also put your name under that.
 Now create a triangle using side c, 1, and 2. Again order matters, copy side c and then place 1 at
the one of the ends of side c… then use 2 to close the triangle.
 Compare your answer with the person beside you. Compare it with 4 people around you…
 Repeat these steps but with side b, 3, and 4
 Discuss with your partner: Do you think SAA is a congruence criteria? Why or why not?
8. Given an Angle, a Side, and a Side. (ASS) (this is the same as SSA just the reverse order)
 Grab a piece of patty paper; in a corner write the criteria that you are currently using to create the
triangle. So in this case put (ASS), and then put your name under that.
 Now create a triangle using 2, and side c, and then side b. This one is a little tricky to form so let me
try to help you… Copy 2, then use side c as one of the sides of 2 and then connect side b to other
endpoint of side c so that it closes the triangle.
 Compare your answer with the person beside you. Compare it with 4 people around you…
 Discuss with your partner: Do you think ASS is a congruence criteria? Why or why not?
 Repeat these steps but with 1, and side a, and then side e.
 Compare your answer with the person beside you. Compare it with 4 people around you…
 Discuss with your partner: Do you think ASS is a congruence criteria? Why or why not?
9. Given a Side, a Side, and a Side. (SSS)
 Grab a piece of patty paper; in a corner write the criteria that you use to create the triangle. So in this
case put (SSS), and then put your name under that.
 Now create a triangle using side a, and side b, and then side c. This one is a little tricky to form so let
me try to help you… Copy side a onto a piece of patty paper, then copy side b onto a different piece of
patty paper, and then copy c onto a different piece of patty paper… now overlap them until they close
the triangle… copy that triangle onto one of those sheets.
 Compare your answer with the person beside you. Compare it with 4 people around you…
 Repeat these steps but with side b, and side c, and then side d
 Compare your answer with the person beside you. Compare it with 4 people around you…
 Discuss with your partner: Do you think this is a congruence criteria? Why or why not?
10. Given an Angle, an Angle, and an Angle. (AAA)
 Grab a piece of patty paper; in a corner write the criteria that you use to create the triangle. So in this
case put (AAA), and then also put your name under that.
 Now create a triangle using 1, 3, and 5. This one is a little tricky to form so let me try to help
you… Copy 1 onto a piece of patty paper, then copy 3 onto a different piece of patty paper, and then
 5 onto a different piece of patty paper… now overlap them until they close the triangle… copy that
triangle onto one of those sheets.
 Compare your answer with the person beside you. Compare it with 4 people around you…
 Do you think this is a congruence criteria? Why or why not?
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Geometry
Triangle Congruence Criteria
Unit CO.8
ON A SEPARATE PIECE OF PAPER (YOUR NOTES) SUMMARIZE YOUR FINDINGS BY
COMPLETING THE FOLLOWING:
1.
S
was / was not
a congruence criteria
2.
A
was / was not
a congruence criteria
3.
SS
was / was not
a congruence criteria
4.
AA
was / was not
a congruence criteria
5.
ASA was / was not
a congruence criteria
6.
SAS was / was not
a congruence criteria
7.
SAA was / was not
a congruence criteria
8.
ASS was / was not
a congruence criteria
9.
SSS was / was not
a congruence criteria
10.
AAA was / was not
a congruence criteria
MISCONCEPTIONS (What are the typical errors or difficult areas? Also suggest ways to teach them.)
 Students tend to forget that angle and segment correspondence is critical for determining if the triangles
are congruent.
o Have students determine which two parts are given (either two angles or two sides). Then
determine if the other part is either outside of the two, or in between the two.
 AAA is not criteria for triangle congruence.
 Under certain conditions, ASS may be a congruence postulate. However, for regular geometry students,
ASS will not be used as a congruence postulate. See additional extensions for use with geometry honors
students.
FUTURE CONNECTIONS (What will they use these skills for later?)
 Triangle congruency and transformations will be utilized to discover the characteristics of different
quadrilaterals.
ADDITIONAL EXTENTIONS OR EXPLANATIONS (What needs greater explanation?)
 Students must understand why three congruent corresponding parts is the minimum necessary to prove
congruency.
 Once congruency is established, students must understand corresponding parts of congruent triangles are
congruent.
 Students will be able to explain that AAA and ASS are insufficient to establish triangle congruency.
 (Honors Only) The connection between ASS and Law of Sines should be noted. This will be further
explored through object G.SRT.D.11
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Triangle Congruence Criteria
ASSESSMENT ITEMS (What questions would evaluate these skills?)
Unit CO.8
Determine whether enough information is given to guarantee the two triangles are congruent.
Write SSS for Side-Side-Side
Write SAS for Side-Angle-Side
Write ASA for Angle-Side-Angle
Write AAS for Angle-Angle-Side
Write none if none of these can be used to prove congruence.
SSS
AAS
SAS
None
ASA
AAS
ASA
ASA
SAS
AAS
ASA
None
SSS
SSS
SAS
AAS
ASA
None
Geometry Unit G.CO.8 Triangle Congruence Criteriarev
AAS
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None
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Geometry
Triangle Congruence Criteria
1. Given ∆ABC ≅ ∆ MNO, find the values of x, y and
B
O
Unit CO.8
.
M
37
x = 19
 3x  20 
A
y = 53
y
C
N
From CCSD Geometry Honors Semester 1 Practice Exam 2012 – 2013
For questions 1–3, consider a triangle ABC that has been transformed through rigid motions and its
image compared to XYZ . Determine if the given information is sufficient to draw the provided
conclusion.
1.
Given
Conclusion
A  X
B  Y
C  Z
(A) True
(B) False
2.
Given
A  X
B  Y
BC  YZ
(A) True
(B) False
3.
Given
A   X
AB  XY
ABC  XYZ
Conclusion
ABC  XYZ
Conclusion
ABC  XYZ
BC  YZ
(A) True
(B) False
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