also leads to SSS when the Pythagorean theorem is applied. Have the students explore the situation with a drawing.
They can draw out two congruent right triangles and mark sides so that the triangles have LL. There is already a
congruence guarantee for this, SAS. What would the non-right triangle congruence be for HL? Is this a guarantee?
(It would be SSA, and no, this does not work in triangles that are right.)
Importance of Right Triangles –When using math to model situations that occur in the world around us the right
triangle is used frequently. Have the students think of right angles that they see every day: walls with the ceilings and
the floors, widows, desks, and many more constructed objects. Right triangles are also important in trigonometry
which they will be studying soon. Stressing the usefulness of right triangles will motivate them to think about why
HL guarantees triangle congruence but SSA, in general, does not.
Using Congruent Triangles
The Process –When students first start examining pairs of triangles to determine congruence it is difficult for them
to sort out all the sides and angles.
The first step is for them to copy the figure onto their paper. It is helpful to color code the sides and angles,
congruent sides marked in one color and the congruent angles in another. Some congruent parts will not be marked
in the original figure that is given to the students in the text. For example, there could be an overlapping side that is
congruent to itself, due to the reflexive property; mark it as well. Then they should do a final check to ensure that
the congruent parts do correspond.
The next step is for them to count how many pairs of congruent corresponding sides and how many pairs of congruent
corresponding angles there are. With this information they can eliminate some possibilities from the list of way
to prove triangles congruent. If there is no right angle they can eliminate HL, or if they only have one set of
corresponding congruent angles, they can eliminate both ASA and AAS.
If at this point there is still more than one possibility, they are going to need to decide if an angle is between two
sides or if a side is between two angles. Remind them that both ASA and AAS can be used to guarantee triangle
congruence, and that SAS works, but that SSA can not be used to prove two triangles are congruent.
If all postulates and theorems have been eliminated, then it is not possible to determine if the triangles are congruent.
AAS or SAA –Sometimes students try to list the congruent sides and angles in a circle as they move around the
triangle. This could result in AAS or SAA when there are two pairs of congruent angles and one pair of congruent
sides that is not between the angles. They know AAS proves congruence and want to know if SAA does as well.
When this occurs it is best to redirect their thinking process. With two sets of angles and one set of sides there
are only two possibilities, the side is between the angles or it is another side. When it is between the angles we
have ASA, if it is either of the other two sides we use SAA. This same situation occurs with SSA, but is even more
important since SSA is not a test for congruence. A good way for the students to remember this is that when the
order of SSA is reversed it makes an inappropriate word. This word should not be used in class or in proofs, even if
it is spelled backwards.
Isosceles and Equilateral Triangles
The Useful Definition of Congruent Triangles –The arguments used in the proof of the Base Angle Theorem
apply what the students have learned about triangles and congruent figures in this chapter, and what they learned
about reasoning and implication in the second chapter. It is a lot of information to bring together and students may
need to review before they can fully understand the proof.
They have been practicing with proofs throughout the chapter, so they should be adept with the logic at this point.
2.4. Congruent Triangles