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. 
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? 



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. 

Mark diagram 
TIP: Mark any given
information on your 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 "leftover"
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? 



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? 
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. 
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 
TIP:
Stay openminded. There
may be more than one way to solve a problem. 

Common Parts 
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 openminded. 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.
