Introducing SSS, SAS and ASA Postulates
Unit Title
Congruent Triangles
Subject Area
Mathematics: Geometry
Age Group
14-15 years old
Essential Question
What is the minimum number of statements that
need to be justified before triangle congruence
can be established?
Habit of Mind
Thinking Interdependently: students will be working
and discussing ideas with one another, which will
provide them with access to multiple viewpoints
and may lead to insights the students may not
have developed if they had approached the
topic individually.
Common Core Content
Standards
G-CO8: Explain how the criteria for triangle
congruence (ASA, SAS, and SSS) follow from the
definition of congruence in terms of rigid motions.
State of Ohio Standards
Grade 10 - Geometry and Spatial Sense Standard:
Characteristics and Properties #3
Prerequisite Knowledge
Understanding of rigid transformations, definition
of congruence in terms of rigid transformations
and corresponding parts (sides and angles) with
respect to two congruent triangles.
Learning Objectives
1) Identify the minimum number of corresponding
pairs needed to establish two triangles are
congruent
2) Formulate conjectures that correspond to the
SSS, SAS and ASA Postulates
3) Identify included and non-included angles and
sides
Learning / Lesson Narrative
Allow time for student questions on previous night’s
homework assignment at the start of the period
Entry:
■ Provide each student with string and ask
students to cut three pieces: one 3 inches,
■
■
■
one 4 inches and one 6 inches
Have students form a triangle from the
three pieces and trace the triangle on a
piece of paper using a straightedge
Have students cut out the triangle formed
and compare triangles with the person
sitting next to them. Question: What do you
notice about your triangles? Lead students
back to definition of congruence.
Have students compare their triangle with a
few more students. Once students have
established they are all congruent, ask
students to conjecture whether or not all
the triangles in the class would be
congruent and have them explain their
reasoning out loud.
Body:
Review Discussion (teacher to whole class)
■ Ask students define congruence in terms of
rigid motions (rotation, reflection,
translation) and corresponding parts.
■ Question: According to the definition of
congruence, how many pairs of
corresponding parts do you need to show
two triangles are congruent to prove the
triangles are congruent? - 6. What are the
corresponding parts? 3 pairs of sides and 3
pairs of angles Question: How many parts
do we know are congruent in the triangles
we just constructed? -3 Do we know
anything about the angles? Thus do we
need to prove all six congruence
statements to prove the triangles are
congruent? -No.
Dynamic Geometry Software Investigation
■ What is the minimum number of
corresponding parts we need to show are
congruent to prove the triangles are
■
■
■
■
■
congruent?
As a class, have students come up with a
list of the following groupings of
corresponding parts: one angle, one side,
an angle and a side, two angles, two sides,
three sides, three angles, two angles and
one side, two sides and one angle.
Have students work in pairs to test each of
these groupings on Geometer’s SketchPad
or another form of dynamic geometry
software. Students should construct a
triangle for each case and test if the figure
can be “distorted” to form a noncongruent triangle. Have students create a
document with statements of congruence
or non-congruence with counterexamples
for each case that does not guarantee
triangle congruence.
For students who are unfamiliar with the
software, demonstrate to the class how to
construct a segment with set length and an
angle with set measure. Demonstrate that
non-set lengths and measure can be
adjusted by dragging. Note when the
triangle can be “distorted,” new triangles
are being formed with the same set of
given parameters; thus those parameters
do not guarantee congruence.
Question: what does it mean when the
figure cannot be “distorted”? Only one
triangle can be formed with those
parameters, thus all other triangles with
those parameters must be congruent.
Provide each student with a conjecture
sheet that provides space for the students
to state their findings and draw their
counterexamples. Conjecture statements
such as the following should be provided:
“If _____ ________ of one triangle are
congruent to _____ ________ of another
triangle, then the triangles are ___________.
■
Optional: create spreadsheet on Google
drive and have each pair fill a specific
conjecture as they are working and display
the results to the class as they are working.
Summary/Group Discussion
■ Call on each pair to share one of their
conjectures with the class and draw their
counterexamples on the board if
congruence is not guaranteed. .
■ Be sure to discuss the differences between
ASA and AAS as well as SAS and SSA when
students are presenting. For example, if a
pair states that congruence between two
sides and an angle guarantees
congruence, ask the remaining students if
they all made the same conjecture. If they
did not, have students draw a diagram of
their non-congruent triangles on the board
highlighting the set sides and angles. Ask
the first group to draw a diagram of the
triangle they constructed highlighting the
set sides and angles. Then have the class
determine where the difference between
the two triangle constructions lies and
introduce the terms “included angle” and
“included side.”
Closure:
■ Once finished collecting conjectures,
summarize which “groupings” of
corresponding parts showed congruence:
SSS, SAS, ASA and AAS. State that SSS, SAS,
ASA will be assumed as postulates.
■ State that AAS is actually a theorem that
can be proven using one of the three
postulates. Ask students to try to figure out
which postulate can be used for homework
and discuss at the beginning of the next
class.
Also assign students a homework set that consists
of identifying if two triangles are congruent based
on a diagram and given information: Classroom
Exercise p. 123 (all) Written Exercises p. 124 1-17
(odd)
Assessment
Instructional Materials and
Resources
Technology Requirements

Informally assess the students’ responses to
questions during the discussion portion
specifically looking to see if they can recall
and use terms such as congruence and
rigid transformations

Have students save, print or e-mail their
activity from the Sketchpad investigation

Monitor students while working on the
activity to see if they are dragging their
figures to check for all possibilities and to
check if they are including all the set
information

Monitor student responses on the excel
worksheet and conjecture sheet while
student are working

Informally assess how well students can
explain and defend their conjectures
during the presentation portion of the
lesson

Review the students’ individual work on
homework assignment next lesson
Each student will need
■ String
■ Ruler/Straightedge
■ Scissors
■ Paper
■ handout with guided conjectures and
space for other findings to be written
■ homework handout (may be assigned
problems from textbook)
■
Access to dynamic geometry software such
■
as GeoGebra or Geometers Sketchpad
Student and teacher access to Google
Drive (optional)
Strategies for Diverse Learners
Students will be working with multiple modalities
throughout the lesson. Students will have access to
written, spoken and visual representations of the
material. Students may also be strategically paired
so that students with dissimilar weaknesses may
work together to aid each other in learning and
concept development.
Reflection
(after lesson has been taught)
This lesson may be adjusted by having students
complete the initial activity by constructing
triangles with a ruler and given measurements as
opposed to string constructions. This will conserve
time so students may spend more time analyzing
the data and formulating their conjectures. This
lesson may also be improved by beginning with a
discussion of “included” sides and angles, so that
students may include examples for each of those
specific cases in their investigation.