CK-12 Geometry
Page 16
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 and 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?
A. the measures of and
B. the measures of and
C. the measures of and
D. the measures of sides and
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 , that side is . In triangle , the corresponding side is . 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.
In the figure, is NOT congruent to even though they have two pairs of congruent sides and a pair of congruent angles. 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 Obtuse triangles have one angle measuring between and Equilateral triangles have congruent sides, and all angles measure Right triangles have one angle measuring exactly
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.
Example 3
Which side of right triangle is the hypotenuse?
Looking at , you can identify 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 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 and and hypotenuse measuring units, the following equation is true.
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.
Example 4
What information would you need to prove that these two triangles were congruent using the HL theorem?
A. the measures of sides and
B. the measures of sides and
C. the measures of angles and
D. the measures of angles and
Since these are right triangles, you only need one leg and the hypotenuse to prove congruence. Legs and are congruent, so you need to find the lengths of the hypotenuses. The hypotenuse of is . The hypotenuse of is . So, you need to find the measures of sides and . 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.
Complete the following congruence statement, if possible _________.
What postulate allows you to make the congruence statement in 1, or, if it is not possible to make a congruence statement explain why.
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 .
Complete the following congruence statement, if possible _________.
What postulate allows you to make the congruence statement in 4, or, if it is not possible to make a congruence statement explain why.
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.
Complete the following congruence statement, if possible ________.
What postulate allows you to make the congruence statement in 7, or, if it is not possible to make a congruence statement explain why.
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?
Write one or two sentences and use a diagram to show why AAA is not a triangle congruence postulate.
Do the following proof using a two-column format.
Given: and intersect at , and
Prove:
Statement Reason
1.
1. Given
2. (Finish the proof using more steps!)
2.
Review Answers
HL triangle congruence postulate
and
SAS triangle congruence postulate
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:
These triangles are not congruent.
Statement Reason
1.
1. Given
2.
2. Given
3. and intersect at
3. Given
4. and are vertical angles
4. Definition of vertical angles
5.
5. Vertical angles theorem
6.
6. SAS triangle congruence postulate
7.
7. Definition of congruent triangles (CPCTC)
Using Congruent Triangles
Learning Objectives
Apply various triangles congruence postulates and theorems.
Know the ways in which you can prove parts of a triangle congruent.
Find distances using congruent triangles.
Use construction techniques to create congruent triangles.
Introduction
As you can see, there are many different ways to prove that two triangles are congruent. It is important to know all of the different way that can prove congruence, and it is important to know which combinations of sides and angles do not prove congruence. When you prove properties of polygons in later chapters you will frequently use
Congruence Theorem Review
As you have studied in the previous lessons, there are five theorems and postulates that provide different ways in which you can prove two triangles congruent without checking all of the angles and all of the sides. It is important to know these five rules well so that you can use them in practical applications.
Name Corresponding congruent parts Does it prove congruence?
SSS Three sides Yes
SAS Two sides and the angle between them Yes
ASA Two angles and the side between them Yes
AAS Two angles and a side not between them Yes
HL A hypotenuse and a leg in a right triangle Yes
AAA Three angles No—it will create a similar triangle, but not of the same size
SSA Two sides and an angle not between them No—this can create more than one distinct triangle
When in doubt, think about the models we created. If you can construct only one possible triangle given the constraints, then you can prove congruence. If you can create more than one triangle within the given information, you cannot prove congruence.
Example 1
What rule can prove that the triangles below are congruent?
A. SSS
B. SSA
C. ASA
D. AAS
The two triangles in the picture have two pairs of congruent angles and one pair of corresponding congruent sides. So, the triangle congruence postulate you choose must have two (for the angles) and one (for the side). You can eliminate choices and for this reason. Now that you are deciding between choices and , you need to identify where the side is located in relation to the given angles. It is adjacent to one angle, but it is not in between them. Therefore, you can prove congruence using AAS. The correct answer is D.
Proving Parts Congruent
It is one thing to identify congruence when all of the important identifying information is provided, but sometimes you will have to identify congruent parts on your own. You have already practiced this in a few different ways. When you were testing SSS congruence, you used the distance formula to find the lengths of sides on a coordinate grid. As a review, the distance formula is shown below.
You can use the distance formula whenever you are examining shapes on a coordinate grid.
When you were creating two-column and flow proofs, you also used the reflexive property of congruence. This property states that any segment or angle is congruent to itself. While this may sound obvious, it can be very helpful in proofs, as you saw in those examples.
You may be tempted to use your ruler and protractor to check whether two triangles are congruent. However, this method does not necessarily work because not all pictures are drawn to scale.
Example 2
How could you prove in the diagram below?
We can already see that and . We may be able to use SSS or SAS to show the triangles are congruent. However, to use SSS, we would need and there is no obvious way to prove this. Can we show that two of the angles are congruent? Notice that and are vertical angles (nonadjacent angles made by the intersection of two lines—i.e., angles on the opposite sides of the intersection).
The Vertical Angle Theorem states that all vertical angles are also congruent. So, this tells us that . Finally, by putting all the information together, you can confirm that by the SAS Postulate.
Finding Distances
One way to use congruent triangles is to help you find distances in real life—usually using a map or a diagram as a model.
When using congruent triangles to identify distances, be sure you always match up corresponding sides. The most common error.png on this type of problem involves matching two sides that are not corresponding.
Example 3
The map below shows five different towns. The town of Meridian was given its name because it lies exactly halfway between two pairs of cities: Camden and Grenata, and Lowell and Morsetown.
Using the information in the map, what is the distance between Camden and Lowell?
The first step in this problem is to identify whether or not the marked triangles are congruent. Since you know that the distance from Camden to Meridian is the same as Meridian to Grenata, those two sides are congruent. Similarly, since the distance from Lowell to Meridian is the same as Meridian to Morsetown, those two sides are also a congruent pair. The angles between these lines are also congruent because they are vertical angles.
So, by the SAS postulate, these two triangles are congruent. This allows us
to find the distance between Camden and Lowell by identifying its corresponding side on the other triangle. Because they are both opposite the vertical angle, the side connecting Camden and Lowell corresponds to the side connecting Morsetown and Grenata. Since the triangles are congruent, these corresponding sides will also be congruent to each other. Therefore, the distance between Camden and Lowell is five miles.
This use of the definition of congruent triangles is one of the most powerful tools you will use in geometry class. It is often abbreviated as CPCTC, meaning Corresponding Parts of Congruent Triangles are Congruent.
Constructions
Another important part of geometry is creating geometric figures through construction. A construction is a drawing that is made using only a straightedge and a compass—you can think of construction as a special game in geometry in which we make figures using only these tools. You may be surprised how many shapes can be made this way.
Example 4
Use a compass and straightedge to construct the perpendicular bisector of the segment below.
Begin by using your compass to create an arc with the same distance from a point as the segment.
Repeat this process on the opposite side.
Now draw a line through the two points of intersections. This forms the perpendicular bisector.