The following examples come from RegentsPrep.org
Practice Problems:
Congruence – Congruent

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.




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:


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.


So, why do other combinations not work?

Once you prove your triangles are congruent, the “leftover” 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 “followup” theorem to be used AFTER the triangles are known to be 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. 
Remember to look for ONLY these combinations for congruent triangles: SAS, ASA, SSS, AAS, and HL(right triangle) 

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 
Answer SAS congruent to SAS 


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 
Answer ASA congruent to ASA 

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 
Answer SAS congruent to SAS The vertical angles are congruent. 

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 
Answer ASA congruent to ASA 

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.


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.
A transversal is a line that intersects two or more lines (in the same plane). When lines intersect, angles are formed in several locations. Certain angles are given “names” that describe “where” the angles are located in relation to the lines. These names describe angles whether the lines involved are parallel or not parallel.
Remember that:
– the word INTERIOR means BETWEEN the lines.
– the word EXTERIOR means OUTSIDE the lines.
– the word ALTERNATE means “alternating sides” of the transversal.
When the lines are NOT parallel … 
When the lines are parallel… 
The names “alternate interior angles”, “alternate exterior angles”, “corresponding angles”, and “interior angles on the same side of the transversal” are used to describe specific angles formed when lines intersect. These names are used both when lines are parallel and when lines are not parallel. Let’s examine these angles, and other angles, when the lines are parallel. 
Look carefully at the diagram below: 
Hint: If you draw a Z on the diagram, the alternate interior angles
are found in the corners of the Z. The Z may also be a backward Z.


Look carefully at the diagram below: 



Hint: If you took a picture of one corresponding angle and slid
the angle up (or down) the same side of the transversal, you
would arrive at the other corresponding angle.
Also: If you draw an F on the diagram, the corresponding angles can be
found in the “corners” of the F. The F may be backward and/or upsidedown.





Of course, there are also other angle relationships occurring when working with parallel lines.
Refresh your memory using the diagram below: 

This is an “old” idea about angles revisited. Since a 
