HCPPS Worthwhile Geometry Task
Bulletin Board Congruence
Adapted from Discovering Geometry by Michael Serra
Common Core Standard
G.CO.8: Explain how the criteria for triangle congruence (ASA, SAS, SAA, and SSS) follow
from the definition of congruence in terms of rigid motions.
Common Core Traditional Pathway: Geometry, Unit 1
The Task
Tony and Maya are creating triangular decorations for the bulletin boards at school. They want
all of the triangles to look identical. The two students need to enlist some friends to help, but
they must work quickly to finish the project on time. Given that each triangle will have 3 angle
measurements and 3 side measurements, what is the least amount of information they need to
provide to their friends so that only identical triangles are formed?
Facilitator Notes
1. Prior to this task, review the definition of congruence and how to set up correspondence
between points, sides, and angles of polygons. Discuss possible shortcuts to set up
triangle congruence.
2. Six stations are set up around the room that will provide students with different
information about the triangular decoration, ABC. Materials necessary at each station
include notebook paper, patty paper, protractors, and rulers.
3. Place students in rotating groups that will be given approximately 10 minutes at each
station. Their task is to determine if the information provided is enough to ensure that all
triangles created with the given criteria will result in identical triangles for the bulletin
boards.
4. The stations are set up as follows:
*Station One- If only given AB  14.7 cm, AC  18 cm, and  A  34 , will
all triangles created be the same?
*Station Two- If only given  A  34 , AB  14.7 cm, and  B  92 , will
all triangles created be the same?



*Station Three- If only given AB  14.7 cm,  B  92 , and  C  54 , will
all triangles created
be the same?


*Station Four- If only given  A  34 ,  B  92 , and  C  54 , will
all triangles created be the same? 


*Station Five- If only given AB  14.7 cm, AC  18 cm, and BC  10 cm, will
all triangles createdbe the same?


*Station Six- If only given AB  14.7 cm, AC  18 cm, and  C  54 , will
all triangles created be the same?



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
information on this license, refer 
to: http://creativecommons.org/licenses/by-nc/3.0/.

HCPPS Worthwhile Geometry Task
5. Allow students to investigate each station using the constructions tools and given
parameters. Encourage students to make as many sketches as possible to come up with
varying examples for each scenario. Once groups have discussed their findings, instruct
students to record their observations on the recording sheet and end each station by
forming a final conjecture outlining their results.
Follow-Up Questions
1. Is one side or one angle enough information to create triangle congruence? What about
two pieces of information?
2. Which stations resulted in the creation of at least one triangle of a different size or shape
while still utilizing the given criteria?
3. Which stations resulted in the creation of all identical triangles no matter how many
different ways you put together the given dimensions and/or angle measures?
4. Did it seem to matter where a given angle or side measurement was located in relation to
the other given measurements? Did its location affect your ability to create an identical
triangle?
5. What conjectures can be formed based on the observations recorded by your group?
Extension Activities
1. Use a geometry computer program or other tools to test the following variation of the
SSA case: If two sides and the angle opposite the longer of the two sides in one triangle
are congruent to two sides and the corresponding angle in another triangle, then the
triangles are congruent. Can you find a counterexample?
2. Is there a conjecture similar to the SSS Congruence Conjecture that you can make about
congruence between quadrilaterals? For example, is SSSS a shortcut for quadrilateral
congruence? Or, if three sides and a diagonal of one quadrilateral are congruent to the
corresponding three sides and diagonal of another quadrilateral, must the two
quadrilaterals be congruent (SSSD)? Investigate. Write a paragraph explaining how your
conjectures follow from the triangle congruence conjectures you’ve learned. (Page 250251, Discovering Geometry)
Solutions
Triangle used for this task: [Insert Figure A]
*Station One- Given AB  14.7 cm, AC  18 cm, and  A  34 
Students will create different options for a triangle with these parameters, but will
discover that only identical triangles will result. Triangles may be transformed (translated,
rotated, reflected), but all triangles will be congruent.
Students will form the conjecture that if



two sides and the angle between them (included angle) in one triangle are congruent to two sides
and the included angle in another triangle, then the triangles are congruent. (SAS Congruence)
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HCPPS Worthwhile Geometry Task
*Station Two- Given  A  34 , AB  14.7 cm, and  B  92 
Students will create different options for a triangle with these parameters, but will
discover that only identical triangles will result. Triangles may be transformed (translated,
rotated, reflected),
but all triangles will be congruent.
Students will form the conjecture that if



two angles and the side between them (included side) in one triangle are congruent to two angles
and the included side in another triangle, then the triangles are congruent. (ASA Congruence)
*Station Three- Given AB  14.7 cm,  B  92 , and  C  54 
Students will create different options for a triangle with these parameters, but will
discover that only identical triangles will result. Triangles may be transformed (translated,
rotated, reflected), but all triangles
will be congruent.
Students will form the conjecture that if



two angles and the side not between them (non-included side) in one triangle are congruent to
two angles and the corresponding side not between them in another triangle, then the triangles
are congruent. (SAA Congruence)
*Station Four- Given  A  34 ,  B  92 , and  C  54 
Students will create different options for a triangle with these parameters and will
discover it is possible for non-congruent triangles to result. Triangles will always be similar to
each other (same shape), however students will create triangles of varying sizes. Students will



form the conjecture that if three angles in one triangle are congruent to three angles in another
triangle, then the triangles are not necessarily congruent. (AAA)
*Station Five- Given AB  14.7 cm, AC  18 cm, and BC  10 cm
Students will create different options for a triangle with these parameters, but will
discover that only identical triangles will result. Triangles may be transformed (translated,
rotated, reflected), but all triangles will be congruent. Students will form the conjecture that if



three sides of one triangle are congruent to three sides of another triangle, then the triangles are
congruent. (SSS Congruence)
*Station Six- Given AB  14.7 cm, AC  18 cm, and  C  54 
Students will create different options for a triangle with these parameters and will
discover it is possible for non-congruent triangles to result. Students will form the conjecture that
if two sides and an angle that is not included in one triangle are congruent to the corresponding



two sides and an angle that is not included in another triangle, then the two triangles are not
necessarily congruent. (SSA)
Follow Up Questions (Sample Answers):
1. Is one side or one angle enough information to create triangle congruence? What about
two pieces of information?
 Give students sample measurements to test these conjectures. For example,
provide students a side length of 4 inches and require them to come up with as
many possible triangles that each contain at least one side measuring 4 inches.
Students will discover that many non-congruent triangles will result. Next,
provide students two pieces of information. For example, one side measuring 2.5
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HCPPS Worthwhile Geometry Task
inches and one angle measuring 50. Other possibilities include two side
measurements or two angle measurements.
2. Which stations resulted in the creation of at least one triangle of a different size or shape
while still utilizing the given criteria?
 Stations four and six
3. Which stations resulted in the creation of all identical triangles no matter how many
different ways you put together the given dimensions and/or angle measures?
 Stations one, two, three, and five
4. Did it seem to matter where a given angle or side measurement was located in relation to
the other given measurements? Did its location affect your ability to create an identical
triangle?
 Discuss the vocabulary “included angle” or “included side.” Discuss the role of an
included angle and how it affects the side located directly opposite it or across
from it in a triangle. Students should understand why SAS does result in
congruence and SSA does not. Use AngLegs or a Geometry computer program
like Sketchpad to model.
5. What conjectures can be formed based on the observations recorded by your group?
 SSS, SAS, SAA, and ASA Congruence
 AAA, SSA does not necessarily result in triangle congruence
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information on this license, refer to: http://creativecommons.org/licenses/by-nc/3.0/.
HCPPS Worthwhile Geometry Task
Name: ____________________________
Date: ________________
Bulletin Board Congruence Investigation
Station #
Given Information
Conjecture
Diagram
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information on this license, refer to: http://creativecommons.org/licenses/by-nc/3.0/.