Download Project Congruence G.CO.7 Use the definition of congruence in

Project Congruence
G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles
are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are
congruent.
G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the
definition of congruence in terms of rigid motions.
G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent;
when a transversal crosses parallel lines, alternate interior angles are congruent and
corresponding angles are congruent; points on a perpendicular bisector of a line segment are
exactly those equidistant from the segment’s endpoints.
G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent,
opposite angles are congruent, the diagonals of a parallelogram bisect each other, and
conversely, rectangles are parallelograms with congruent diagonals.
G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass
and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing
perpendicular lines, including the perpendicular bisector of a line segment; and constructing a
line parallel to a given line through a point not on the line.
A. Create a unique design that meets the following criteria:
1. Begin with a parallelogram and subdivide so that the rest of the criteria hold.
2. It has at least 3 parallel line segments
3. It makes use of midpoints of segments
4. It consists only of triangles, though these may fill the interior of other shapes.
5. At least three of the triangles are congruent.
6. It has at least two other triangles that are congruent to each other but not to the set of
three congruent triangles. This set can also not be similar to the set of three congruent
triangles.
7. Make a list of families of congruent triangles.
8. Image must have at least one set of similar triangles and make a list of families of similar
triangles.
B. Once you have your design, make a second copy and mark up known information such as
line segments you know are parallel and congruent segments coming from the definition of
the shape or from using the midpoints. Get your design checked by the instructor before
moving on to part C.
1. Sketch your design on paper first.
2. Construct your design using GeoGebra. Use the tools to construct parallels,
perpendiculars, midpoints, do not freehand draw them. The rigid motion tools can
also help with constructing your design.
3. Export final design as a png. Download and print several copies. Or do a capture
and copy into a word document.
C. Proving triangle congruence. You will be doing 4 total proofs, 2 with rigid motions and 2
without using rigid motions but making use of SSS, SAS, ASA.
1. Pick 2 of your triangles from criteria 5. Justify triangle congruence using rigid motions.
And justify triangle congruence without. (2 proofs)
2. Pick 2 of your triangle form criteria 6. Justify triangle congruence using rigid motions.
And Justify triangle congruence using SSS, SAS, or ASA. (2 proofs).
Example of a design that works. This one is not available for your use.
Rubrics
Rubric 1: for Part A.
Math Practice standard 6 – attends to precision and G.C0.12.
Exceeding
All 8 criteria from part
A are met.
Meeting
6 criteria from part A
are met, including
criteria 1.
Approaching
Criteria 1 and at least 3
others are met.
Beginning
Criteria 1 from part A is
met, plus 1 other.
Rubric for the proofs, G.CO.7, 9, and 11.
Characteristics
of Proof
Prerequisite
information or
assumptions
Beginning
Approaching
Meeting
Exceeding
Missing all needed
prerequisite
information or
assumptions.
Missing some
prerequisite
information or
some assumptions
are not stated.
Prerequisite
information or
assumptions are
stated.
Logical Steps
There are little to
no logical steps
evident.
Some logical steps
are missing OR
Some logical steps
are incorrect.
Necessary
Information
There is little to no
necessary
information.
The proof is
missing some
necessary
information.
Overall
Structure
The proof does not
work and needs
major work.
The proof does not
work, but is close.
Enough logical
steps are provided
so the reader can
make sense that
the proof is
correct and
complete.
Sufficient and
necessary
information. Some
irrelevant
information may
be included.
The proofs works,
but could be made
cogent and solid.
The minimum
amount of
prerequisite
information or
assumptions are
stated.
Logical steps are
clear and connect
in a flow or
sequence.
Sufficient and
necessary
information. No
excess irrelevant
information is
included.
The proof is
elegant and
powerful.
Rubric for Professionalism
Exceeding
Meeting plus student
adjusts work based on
feedback from teacher
and peers on multiple
occasions.
Meeting
Timely –
Student turns in all the
work on time.
Neat –
Clean and labeled
design, proofs are
clearly written.
Approaching
Mostly on time –
Student turns in most
of the work on time.
Mostly neat – labels
where needed on
design. Proofs are
mostly clearly written.
Beginning
Nearly half (or more) of
the work is not done
on time.
Design is poorly labeled
or hard to make out,
proofs are not written
clearly.