Download 5 blog notes for congruent triangle proofs

Theorems for Congruent
Triangles
Latest news bulletin:
The most popular congruent figures are triangles!
In many geometrical proofs, it may be necessary to prove that
two triangles are congruent to each other. The task may simply
be to prove the triangles congruent, or it may be to use these
congruent triangles to gain additional information.
When triangles are congruent and one triangle is placed on top of the other,
the sides and angles that coincide (are in the same positions) are called corresponding parts.
Example:
When two triangles are congruent, there are
6 facts that are true about the triangles:


the triangles have 3 sets of congruent
(of equal length) sides and
the triangles have 3 sets of congruent
(of equal measure) angles.
NOTE: The corresponding
congruent sides are marked
with small straight line
segments called hash marks.
The corresponding congruent
angles are marked with arcs.
The 6 facts for our congruent triangles example:
Note:
The order of the letters in the names of the triangles should display the corresponding
relationships. By doing so, even without a picture, you would know that <A would be
congruent to <D, and
would be congruent to
, because they are in the same position in
each triangle name.
Wow! Six facts for every set of congruent triangles!
Fortunately, when we need to PROVE (or show) that triangles are congruent, we do NOT need
to show all six facts are true. There are certain combinations of the facts that are sufficient to
prove that triangles are congruent. These combinations of facts guarantee that if a triangle can
be drawn with this information, it will take on only one shape. Only one unique triangle can be
created, thus guaranteeing that triangles created with this method are congruent.
Methods for Proving (Showing) Triangles to be Congruent
SSS
If three sides of one triangle are congruent to three sides of
another triangle, the triangles are congruent.
(For this method, the sum of the lengths of any two sides must be greater than
the length of the third side, to guarantee a triangle exists.)
SAS
If two sides and the included angle of one triangle are
congruent to the corresponding parts of another triangle,
the triangles are congruent. (The included angle is the angle formed
by the sides being used.)
ASA
If two angles and the included side of one triangle are
congruent to the corresponding parts of another triangle,
the triangles are congruent. (The included side is the side between
the angles being used. It is the side where the rays of the angles would
overlap.)
AAS
If two angles and the non-included side of one triangle are
congruent to the corresponding parts of another triangle,
the triangles are congruent. (The non-included side can be either of
the two sides that are not between the two angles being used.)
HL
Right
Triangles
Only
If the hypotenuse and leg of one right triangle are
congruent to the corresponding parts of another right
triangle, the right triangles are congruent. (Either leg of the right
triangle may be used as long as the corresponding legs are used.)
BE CAREFUL!!!
Only the combinations
listed above will give
congruent triangles.
So, why do other combinations not work?
Methods that DO
AAA
NOT Prove Triangles to be Congruent
AAA works fine to show that triangles are the
same SHAPE (similar), but does NOT work to also
show they are the same size, thus congruent!
Consider the example at the right.
SSA
or
ASS
SSA (or ASS) is humorously referred to as
the "Donkey Theorem".
This is NOT a universal method to prove triangles
congruent because it cannot guarantee that one
unique triangle will be drawn!!
You can easily draw 2 equilateral
triangles that are the same shape
but are not congruent (the same
size).
The SSA (or ASS) combination affords the possibility of creating
zero, one, or two triangles. Consider this diagram of triangle
DEF. If for the second side, EF is equal to EG (the minimum
distance needed to create a triangle), only one triangle can be drawn.
However, if EF is greater than EG, two triangles can be drawn as
shown by the dotted segment. Should EF be less than the
minimum length needed to create a triangle, EG, no triangle can
be drawn.
The possible "swing" of side
can create two different triangles which causes our
problem with this method. The first triangle, below, and the last triangle both show
SSA, but they are not congruent triangles.
The combination of SSA (or ASS) creates a unique triangle ONLY when working in a
right triangle with the hypotenuse and a leg. This application is given the name HL
(Hypotenuse-Leg) for Right Triangles to avoid confusion. You should not list SSA
(or ASS) as a reason when writing a proof.
Once you prove your triangles are congruent, the "left-over" pieces that
were not used in your method of proof, are also congruent.
Remember, congruent triangles have 6 sets of congruent pieces. We
now have a "follow-up" theorem to be used AFTER the triangles are
known to be congruent:
Theorem: (CPCTC) Corresponding parts
of congruent triangles are congruent.