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Proof Using SAS and HL
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C ONCEPT
1
1
Proof Using SAS and HL
Learning Objectives
•
•
•
•
Understand and apply the SAS Congruence Postulate.
Identify the distinct characteristics and properties of right triangles.
Understand and apply the HL Congruence Theorem.
Understand that SSA does not necessarily prove triangles are congruent.
Introduction
You have already seen three different ways to prove that two triangles are congruent (without measuring six angles
and six sides). Since triangle congruence plays such an important role in geometry, it is important to know all of
the different theorems and postulates that can prove congruence, and it is important to know which combinations of
sides and angles do not prove congruence.
SAS Congruence
By now, you are very familiar with postulates and theorems using the letters S and A to represent triangle sides and
angles. One more way to show two triangles are congruent is by the SAS Congruence Postulate.
SAS Triangle Congruence Postulate: If two sides and the included angle in one triangle are congruent to two sides
and the included angle in another triangle, then the two triangles are congruent.
Like ASA and AAS congruence, the order of the letters is very significant. You must have the angles between the
two sides for the SAS postulate to be valid.
Once again you can test this postulate using physical models (such as pieces of uncooked spaghetti) for the sides of
a triangle. You’ll find that if you make two pairs of congruent sides, and lay them out with the same included angle
then the third side will be determined.
Example 1
What information would you need to prove that these two triangles were congruent using the SAS postulate?
Concept 1. Proof Using SAS and HL
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A. the measures of 6 HJG and 6 ST R
B. the measures of 6 HGJ and 6 SRT
C. the measures of HJ and ST
D. the measures of sides GJ and RT
If you are to use the SAS postulate to establish congruence, you need to have the measures of two sides and the
angle in between them for both triangles. So far, you have one side and one angle. So, you must use the other side
adjacent to the same angle. In 4GHJ, that side is HJ. In triangle 4RST , the corresponding side is ST . So, the
correct answer is C.
AAA and SSA relationships
You have learned so many different ways to prove congruence between two triangles, it may be tempting to think that
if you have any pairs of congruent three elements (combining sides or angles), you can prove triangle congruence.
However, you may have already guessed that AAA congruence does not work. Even if all of the angles are equal
between two triangles, the triangles may be of different scales. So, AAA can only prove similarity, not congruence.
SSA relationships do not necessarily prove congruence either. Get your spaghetti and protractors back on your desk
to try the following experiment. Choose two pieces of spaghetti at given length. Select a measure for an angle that
is not between the two sides. If you keep that angle constant, you may be able to make two different triangles. As
the angle in between the two given sides grows, so does the side opposite it. In other words, if you have two sides
and an angle that is not between them, you cannot prove congruence.
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In the figure, 4ABC is NOT congruent to 4FEH even though they have two pairs of congruent sides and a pair of
congruent angles. FG ∼
= FH ∼
= AC and you can see that there are two possible triangles that can be made using this
combination SSA.
Example 2
Can you prove that the two triangles below are congruent?
Note: Figure is not to scale.
The two triangles above look congruent, but are labeled, so you cannot assume that how they look means that they
are congruent. There are two sides labeled congruent, as well as one angle. Since the angle is not between the two
sides, however, this is a case of SSA. You cannot prove that these two triangles are congruent. Also, it is important
to note that although two of the angles appear to be right angles, they are not marked that way, so you cannot assume
that they are right angles.
Right Triangles
So far, the congruence postulates we have examined work on any triangle you can imagine. As you know, there
are a number of types of triangles. Acute triangles have all angles measuring less than 90◦ .Obtuse triangles have
one angle measuring between 90◦ and 180◦ .Equilateral triangles have congruent sides, and all angles measure
60◦ .Right triangles have one angle measuring exactly 90◦ .
In right triangles, the sides have special names. The two sides adjacent to the right angle are called legs and the side
opposite the right angle is called the hypotenuse.
Concept 1. Proof Using SAS and HL
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Example 3
Which side of right triangle BCDis the hypotenuse?
Looking at 4BCD, you can identify 6 CBD as a right angle (remember the little square tells us the angle is a right
angle). By definition, the hypotenuse of a right triangle is opposite the right angle. So, side CD is the hypotenuse.
HL Congruence
There is one special case when SSA does prove that two triangles are congruent-When the triangles you are
comparing are right triangles. In any two right triangles you know that they have at least one pair of congruent
angles, the right angles.
Though you will learn more about it later, there is a special property of right triangles referred to as the Pythagorean
theorem. It isn’t important for you to be able to fully understand and apply this theorem in this context, but it is
helpful to know what it is. The Pythagorean Theorem states that for any right triangle with legs that measure a and
b and hypotenuse measuring c units, the following equation is true.
a2 + b2 = c2
In other words, if you know the lengths of two sides of a right triangle, then the length of the third side can be
determined using the equation. This is similar in theory to how the Triangle Sum Theorem relates angles. You know
that if you have two angles, you can find the third.
Because of the Pythagorean Theorem, if you know the length of the hypotenuse and a leg of a right triangle, you
can calculate the length of the missing leg. Therefore, if the hypotenuse and leg of one right triangle are congruent
to the corresponding parts of another right triangle, you could prove the triangles congruent by the SSS congruence
postulate. So, the last in our list of theorems and postulates proving congruence is called the HL Congruence
Theorem. The “H” and “L” stand for hypotenuse and leg.
HL Congruence Theorem: If the hypotenuse and leg in one right triangle are congruent to the hypotenuse and leg
in another right triangle, then the two triangles are congruent.
The proof of this theorem is omitted because we have not yet proven the Pythagorean Theorem.
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Example 4
What information would you need to prove that these two triangles were congruent using the HL theorem?
A. the measures of sides EF and MN
B. the measures of sides DF and LN
C. the measures of angles 6 DEF and 6 LMN
D. the measures of angles 6 DFE and 6 LNM
Since these are right triangles, you only need one leg and the hypotenuse to prove congruence. Legs DE and LM are
congruent, so you need to find the lengths of the hypotenuses. The hypotenuse of 4DEF is EF. The hypotenuse of
4LMN is MN. So, you need to find the measures of sides EF and MN. The correct answer is A.
Points to Consider
The HL congruence theorem shows that sometimes SSA is sufficient to prove that two triangles are congruent. You
have also seen that sometimes it is not. In trigonometry you will study this in more depth. For now, you might try
playing with objects or you may try using geometric software to explore under which conditions SSA does provide
enough information to infer that two triangles are congruent.
Lesson Summary
In this lesson, we explored triangle sums. Specifically, we have learned:
•
•
•
•
How to understand and apply the SAS Congruence Postulate.
How to identify the distinct characteristics and properties of right triangles.
How to understand and apply the HL Congruence Theorem.
That SSA does not necessarily prove triangles are congruent.
These skills will help you understand issues of congruence involving triangles. Always look for triangles in diagrams, maps, and other mathematical representations.
Review Questions
Use the following diagram for exercises 1-3.
Concept 1. Proof Using SAS and HL
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1. Complete the following congruence statement, if possible 4RGT ∼
= _________.
2. What postulate allows you to make the congruence statement in 1, or, if it is not possible to make a congruence
statement explain why.
3. Given the marked congruent parts in the triangles above, what other congruence statements do you now know
based on your answers to 1 and 2?
Use the following diagram below for exercises 4-6 .
4. Complete the following congruence statement, if possible 4TAR ∼
= _________.
5. What postulate allows you to make the congruence statement in 4, or, if it is not possible to make a congruence
statement explain why.
6. Given the marked congruent parts in the triangles above, what other congruence statements do you now know
based on your answers to 4 and 5?
Use the following diagram below for exercises 7-9.
7. Complete the following congruence statement, if possible 4PET ∼
= ________.
8. What postulate allows you to make the congruence statement in 7, or, if it is not possible to make a congruence
statement explain why.
9. Given the marked congruent parts in the triangles above, what other congruence statements do you now know
based on your answers to 7 and 8?
10. Write one or two sentences and use a diagram to show why AAA is not a triangle congruence postulate.
11. Do the following proof using a two-column format.
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Given: MQ and NP intersect at O.NO ∼
= OQ, and MO ∼
= OP Prove: 6 NMO ∼
= 6 OPN
TABLE 1.1:
Statement
1. NO ∼
= OQ
2. (Finish the proof using more steps!)
Reason
1. Given
2.
Review Answers
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
4RGT ∼
= 4NPU
HL triangle congruence postulate
GR ∼
= PN, 6 T ∼
= 6 U, and 6 R ∼
=6 N
∼
4TAR = 4PIM
SAS triangle congruence postulate
6 T ∼
= 6 P, 6 A ∼
= 6 I, TA ∼
= PI
No triangle congruence statement is possible
SSA is not a valid triangle congruence postulate
No other congruence statements are possible
One counterexample is to consider two equiangular triangles. If AAA were a valid triangle congruence
postulate, than all equiangular (and equilateral) triangles would be congruent. But this is not the case. Below
are two equiangular triangles that are not congruent:
Concept 1. Proof Using SAS and HL
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These triangles are not congruent.
TABLE 1.2:
Statement
1. NO ∼
= OQ
2. MO ∼
= OP
3. MQ and NP intersect at O
4. 6 NOM and 6 QOP are vertical angles
5. 6 NOM ∼
= 6 QOP
6. 4NOM ∼
= 4QOP
∼
6
7. NMO = 6 OPN
Reason
1. Given
2. Given
3. Given
4. Definition of vertical angles
5. Vertical angles theorem
6. SAS triangle congruence postulate
7. Definition of congruent triangles (CPCTC)
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Concept 1. Proof Using SAS and HL