Similarity and Congruence (Scooby-Doo)

scooby-doo-cartoonThe following examples come from RegentsPrep.org

Practice Problems:

Triangles 1

Triangles 2

Angles and Parallel Lines

 

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 andcorresponding 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.

 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 NOT Prove Triangles to be Congruent

AAA

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.

You can easily draw 2 equilateral triangles that are the same shape but are not congruent (the same size).

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!!

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,
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.

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?

             Answer             SAS  congruent to SAS             

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.

 

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



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 “left-over” corresponding pieces.

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

             Answer             ASA congruent to ASA             

 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?

             Answer             SAS congruent to SAS             The vertical angles are congruent.             


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?

             Answer             ASA congruent to ASA             


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 open-minded.  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 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.

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.

When the lines are parallel:
Alternate Interior Angles
(measures are equal
)
The name clearly describes “where”
these angles are located. 

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.
        

Theorem:

If two parallel lines are cut by a transversal, the alternate interior angles are congruent.

 

Theorem:

If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel.

When the lines are parallel:
Alternate Exterior Angles
(measures are equal
)
The name clearly describes “where”
these angles are located.

Look carefully at the diagram below:

Theorem:

If two parallel lines are cut by a transversal, the alternate exterior angles are congruent.

 

Theorem:

If two lines are cut by a transversal and the alternate exterior angles are congruent, the lines are parallel.

When the lines are parallel:
Corresponding Angles

(measures are equal)
Unfortunately, the name of these angles
does not clearly indicate “where” they
are located.  
They are located:
– on the SAME SIDE of the transversal
– one INTERIOR and one EXTERIOR
– and they are NOT adjacent (they don’t touch).
(They lie on the same side of the transversal,
in corresponding positions.)


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 upside-down.

      

Theorem:

If two parallel lines are cut by a transversal, the corresponding angles are congruent.

 

Theorem:

If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel.

When the lines are parallel:
Interior Angles on the Same Side of the Transversal
(measures are supplementary)

Their “name” is simply a description of where the angles are located.

Theorem:

If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary.

 

Theorem:

If two lines are cut by a transversal and the interior angles on the same side of the transversal are supplementary, the lines are parallel.

Of course, there are also other angle relationships occurring when working with parallel lines. 

Vertical Angles
(measures are equal)
Vertical angles are ALWAYS equal, whether you
have parallel lines or not.

   Refresh your memory using the diagram below:

Theorem:

Vertical angles are congruent.

Angles forming a Linear Pair
(Adjacent Angles creating a Straight Line)
(measures are supplementary)

 This is an “old” idea about angles revisited.  Since a
straight angle contains 180°, these two adjacent angles
add to 180.  They form a linear pair.
(
Adjacent angles share a vertex, share a side, and do not overlap.)

Theorem:

If two angles form a linear pair, they are supplementary.
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