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The Concept of Congruence
Congruence - Congruent
You walk into your favorite mall and see dozens
of copies of your favorite CD on sale. All of the
CDs are exactly the same size and shape.
In fact, you can probably think of many objects
that are mass produced to be exactly the same size
and shape.
Objects that are exactly the same size and shape are said to be
congruent.
Congruent objects are duplicates of one another.
If two mathematical figures are congruent and you cut one figure out with a pair of scissors, it will
fit perfectly on top of the other figure.
Mathematicians use the word congruent to describe geometrical
figures.
-- If two quadrilaterals (4 sided) are the same size and shape,
they are congruent.
-- If two pentagons (5 sided) are the same size and shape,
they are congruent.
-- If two polygons (any number of sides) are the same size and
shape,
they are congruent.
-- If two line segments are the same length (they already are the
same
shape), they are congruent.
The mathematical symbol used to denote
congruent is .
The symbol is made up of two parts:
which means the same shape (similar) and
which means the same size (equal).
Congruent
Symbol
When you are looking at congruent figures, be sure to find the sides
and the angles that "match up" (are in the same places) in each
figure. Sides and angles that "match up" are called corresponding
sides and corresponding angles.
In congruent figures, these corresponding parts are also congruent.
The corresponding sides will be equal in measure (length) and that
the corresponding angles will be equal in degrees.
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.
Two triangles are congruent if all pairs of corresponding sides are congruent, and all pairs of
corresponding angles are congruent. Fortunately, we do not need to show all six of these
congruent parts each time we want to show triangles congruent. There are 5 combination
methods that allow us to show triangles to be congruent.
Tips for Working with
Congruent Triangles in Proofs
Remember to look for ONLY these combinations for congruent
triangles:
SAS, ASA, SSS, AAS, and HL(right triangle)
But how do we decide which method we should be using?
Let's look at some examples and tips:
Example 1:
Here is an example problem, using one of the methods mentioned above.
Prove:
:
Which congruent triangle method do you think
is used in this example?
Answ er
Did you notice that the congruent triangle parts
that were given to us were marked up in the
diagram? This technique is very helpful when
trying to decide which method of congruent
triangles to use.
TIP: Mark any given information on your diagram.
Mark diagram
Example 2:
In this example problem, examine the given information, mark the given information on the
diagram as in the first tip, and decide if congruent triangles will help you solve this problem.
Prove:
:
This problem does not ask you to prove the triangles are congruent. This, however, does not mean that you
should not "look" for congruent triangles in this problem. Remember that once two triangles are congruent,
their "left-over" corresponding pieces are also congruent. If you can prove these two triangles are
congruent, you will be able to prove that the segments you need are also congruent since they will be "leftover" corresponding pieces.
Which of the congruent triangle methods
do you think is used in this example?
Answ er
For the triangles in this second example, three sets of corresponding parts were used to prove the
triangles congruent. Can you name the other 3 sets of corresponding parts?
CLICK HERE to see the answer.
Corresponding
Parts
TIP: Look to see if the pieces you need are "parts"
of the triangles that can be proven congruent.
Example 3:
In this example problem, examine the given information, decide what else you need to know, and
then decide the proper method to be used to prove the triangles congruent.
Prove:
There seems to be missing information in this problem. There are only two pieces of congruent information
given. This problem expects you to "find" the additional information you will need to show that the
triangles are congruent. What else do you notice is true in this picture?
Answ er
Which of the congruent triangle methods
do you think is used in this example?
Examine
Diagram
TIP: If not given all needed pieces to prove the
triangles congruent, look to see what else you might
know about the diagram.
Example 4:
In this example problem, examine the given information carefully, mark up the diagram and then
decide upon the proper method to be used to prove the triangles congruent.
When you marked up the diagram, did you mark the information gained from the definition of the angle
bisector? While this problem only gives you two of the three sets of congruent pieces needed to prove the
triangles congruent, it also gives you a "hint" as to how to obtain the third needed set. The "hint" in this
problem is in the form of a definition - the angle bisector.
Answ er
Which of the congruent triangle methods
do you think is used in this example?
Use Definitions
TIP: Know your definitions! If the given
information contains definitions, consider these as
"hints" to the solution and be sure to use them.
This particular example can be solved in more than one way.
Even though the given information gives congruent information about <B and <D, this information is not
needed to prove the triangles congruent. The two triangles in this problem "share" a side (called a common
side). This "sharing" automatically gives you another set of congruent pieces.
More than one
solution
Common Parts
TIP: Stay open-minded. There may be more than
one way to solve a problem.
TIP: Look to see if your triangles "share" parts.
These common parts are automatically one set of
congruent parts.
In summary, when working with congruent triangles, remember to:
1. Mark any given information on your diagram.
2. Look to see if the pieces you need are "parts" of the triangles that can be proven congruent.
3. If not given all needed pieces to prove the triangles congruent, look to see what else you
might know about the diagram.
4. Know your definitions! If the given information contains definitions, consider these as
"hints" to the solution and be sure to use them.
5. Stay open-minded. There may be more than one way to solve a problem.
6. Look to see if your triangles "share" parts. These common parts are automatically one set of
congruent parts.
Remember that proving triangles congruent is like solving a puzzle. Look carefully at the
"puzzle" and use all of your geometrical strategies to arrive at an answer.