Lesson Page

 


  Math B

Theorems for
Beginning Congruent Triangle Proofs

In many proofs, it is necessary to prove that two triangles are congruent to each other.  This task could be the end result of the problem, or the two congruent triangles could be used to further prove corresponding pairs of angles or line segments to be congruent.

Definition:  Two triangles (polygons) are congruent if all pairs of corresponding sides are congruent, and all pairs of corresponding angles are congruent.

In the diagram above,

Since these triangles are congruent,
all corresponding angles are congruent to each other, and
all corresponding sides are congruent to each other. 
 

In the above case, the 6 corresponding parts are:
                      and

 

With the definition in mind, we use the following statement
to signify congruent corresponding parts:
Corresponding Parts of Congruent Triangles are Congruent

Or the shortcut version,
C.P.C.T.C.

 

Good News!! 

In order to show that we have 2 congruent triangles, it is only necessary to show 3 sets of corresponding parts of the triangles are congruent.
 
 

Not-so-Good News!!

 Proofs are fussy on WHICH 3 sets of corresponding parts are congruent when showing triangles to be congruent.


We have five methods to choose from to prove that 2 triangles
are congruent to one another.  Below are the combinations that WILL work.  
 

Methods of 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.

SAS

 If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

ASA

 If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

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.

HL

 If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.

 



Example 1:

Here is a example problem, using one of the methods mentioned above.

The Conclusion is:  :

   
Which of the above methods is used
in this example?
   

Try your hand at matching the corresponding parts for these congruent triangles.
CLICK HERE

 

Did you notice that the triangle parts that were given to us were marked up in the diagram?  This technique is very helpful when deciding which method of congruent triangles to use.
 Mark information on your diagram as it is given.

 



Example 2:

In this example problem, examine the given information, then decide the proper method to be used to prove the triangles congruent. (You may want to draw the
diagram on your own paper, and mark the given congruent parts.)  

The Conclusion is:

   
Which of the congruent triangle
 methods is used in this example?

   

 For the triangles in the second example, 3 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.

 

Remember to look for ONLY these combinations for congruent triangles:
 SAS, ASA, SSS, AAS, and HL

 


           F. Roberts