Investigate congruence by manipulating the parts (sides and angles) of a
triangle. If you can create two different triangles with the same
parts, then those parts do not prove congruence. Can you prove all the
Each triangle congruence theorem uses three elements (sides and angles)
to prove congruence. Select three triangle elements from the top, left
menu to start. (Note: The tool does not allow you to select more
than three elements. If you select the wrong element, simply unselect it
before choosing another element.) This creates those elements in the
On the top of the toolbar, the three elements are listed in order.
For example, if you choose side AB, angle A, and angle B, you will be
working on Angle – Side – Angle. If instead you choose side AB, angle A,
and angle C, you will be working on Angle – Angle – Side. The two
theorems are different, even though both involve two angles and one
Construct your triangle:
Once a triangle is formed with the original three elements, the
triangle move to the bottom, right corner of the work area, and
congruent elements appear. Try to form a second triangle. If the second
triangle can only be formed congruent to the first, then that
arrangement of three elements proves a congruence theorem. If you can
form a non-congruent triangle, then that disproves congruence.
After a second triangle is formed, you will be asked if they are
congruent. You can test congruence by manipulating either triangle.
If the two triangles are congruent, you will be asked if it's possible to make a triangle that is not congruent to the original. If you create a third congruent triangle, you will be given the option to try again.
The Reset button clears the work area and creates new sides and angles for the selected elements.
The New button clears your selection and work area.
For every arrangement of three elements, it is possible to test for
triangle congruence. However, just like two triangle might be mirror
images but still congruent, flipping the order creates identical
possible theorems. For example Angle – Angle – Side is the same as
Side – Angle – Angle because they are the same elements in reverse
How many possibilities are there? Test each of them and organize your results in the table with three columns:
As you work, remember to try every possibility. When creating your
second triangle, see if an element can connect to adjacent elements in
more than one place. Also remember, you may have to turn or flip your
triangles to see how they are congruent. For example, all the triangles
to the right are congruent.
Did all the possibilities work? How are you certain of your
answers? Try comparing your answers to a friend's. If you disagree on
any possible theorem, try re-creating your triangles.
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