CK-12 Geometry
Page 15
Imagine you could pick up without changing its shape and move the whole triangle right and down. If you did this, then points and would be on top of each other, and would also be on top of each other, and and would also coincide.
To analyze the relationship between the points, the distance formula is not necessary. Simply look at how far (and in what direction) the vertices may have moved.
Points and : is and is . is to the right and below .
Points and : is and is . is to the right and below .
Points and : is and is . is to the right and below .
Since the same relationship exists between the vertices, you could move the entire triangle to the right and down. It would exactly cover triangle . These triangles are therefore congruent.
SSS Postulate of Triangle Congruence
The extended example above illustrates that when three sides of one triangle are equal in length to three sides of another, then the triangles are congruent. We did not need to measure the angles—the lengths of the corresponding sides being the same “forced” the corresponding angles to be congruent. This leads us to one of the triangle congruence postulates:
Side-Side-Side (SSS) Triangle Congruence Postulate: If three sides in one triangle are congruent to the three corresponding sides in another triangle, then the triangles are congruent to each other.
This is a postulate so we accept it as true without proof.
You can perform a quick experiment to test this postulate. Cut two pieces of spaghetti (or a straw, or some segment-like thing) exactly the same length. Then cut another set of pieces that are the same length as each other (but not necessarily the same length as the first set). Finally, cut one more pair of pieces of spaghetti that are identical to each other. Separate the pieces into two piles. Each pile should have three pieces of different lengths. Build a triangle with one set and leave it on your desk. Using the other pieces, attempt to make a triangle with a different shape or size by matching the ends. Notice that no matter what you do, you will always end up with congruent triangles (though they might be “flipped over” or rotated). This demonstrates that if you can identify three pairs of congruent sides in two triangles, the two triangles are fully congruent.
Example 2
Write a triangle congruence statement based on the diagram below:
We can see from the tick marks that there are three pairs of corresponding congruent sides: , , and . Matching up the corresponding sides, we can write the congruence statement .
Don’t forget that ORDER MATTERS when writing triangle congruence statements. Here, we lined up the sides with one tic mark, then the sides with two tic marks, and finally the sides with three tic marks.
Lesson Summary
In this lesson, we explored triangle congruence using only the sides. Specifically, we have learned:
How to use the distance formula to analyze triangles on a coordinate grid
How to understand and apply the SSS postulate of triangle congruence.
These skills will help you understand issues of congruence involving triangles, and later you will apply this knowledge to all types of shapes.
Points to Consider
Now that you have been exposed to the SSS Postulate, there are other triangle congruence postulates to explore. The next chapter deals with congruence using a mixture of sides and angles.
Review Questions
If you know that in the diagram below, what are six congruence statements that you also know about the parts of these triangles?
Redraw these triangles using geometric markings to show all congruent parts.
Use the diagram below for exercises 3-7 .
Find the length of each side in
Find the length of each side in
Write a congruence statement relating these two triangles.
Write another equivalent congruence statement for these two triangles.
What postulate guarantees these triangles are congruent?
Exercises 8-10 use the following diagram:
Write a congruence statement for the two triangles in this diagram. What postulate did you use?
Find . Explain how you know your answer.
Find . Explain how you know your answer.
Review Answers
and
One possible answer:
(Note, other answers are possible, but the relative order of the letters does matter.)
SSS
, the side-side-side triangle congruence postulate
. We know this because it corresponds with , so
. Used the triangle sum theorem together with my answer for 9.
Triangle Congruence Using ASA and AAS
Learning Objectives
Understand and apply the ASA Congruence Postulate.
Understand and apply the AAS Congruence Theorem.
Understand and practice two-column proofs.
Understand and practice flow proofs.
Introduction
The SSS Congruence Postulate is one of the ways in which you can prove two triangles are congruent without measuring six angles and six sides. The next two lessons explore other ways in which you can prove triangles congruent using a combination of sides and angles. It is helpful to know all of the different ways you can prove congruence between two triangles, or rule it out if necessary.
ASA Congruence
One of the other ways you can prove congruence between two triangles is the ASA Congruence Postulate. The “S” represents “side,” as it did in the SSS Theorem. “A” stands for “angle” and the order of the letters in the name of the postulate is crucial in this circumstance. To use the ASA postulate to show that two triangles are congruent, you must identify two angles and the side in between them. If the corresponding sides and angles are congruent, the entire triangles are congruent. In formal language, the ASA postulate is this:
Angle-Side-Angle (ASA) Congruence Postulate: If two angles and the included side in one triangle are congruent to two angles and the included side in another triangle, then the two triangles are congruent.
To test out this postulate, you can use a ruler and a protractor to make two congruent triangles. Start by drawing a segment that will be one side of your first triangle and pick two angles whose sum is less than Draw one angle on one side of the segment, and draw the second angle on the other side. Now, repeat the process on another piece of paper, using the same side length and angle measures. What you’ll find is that there is only one possible triangle you could create—the two triangles will be congruent.
Notice also that by picking two of the angles of the triangle, you have determined the measure of the third by the Triangle Sum Theorem. So, in reality, you have defined the whole triangle; you have identified all of the angles in the triangle, and by picking the length of one side, you defined the scale. So, no matter what, if you have two angles, and the side in between them, you have described the whole triangle.
Example 1
What information would you need to prove that these two triangles are congruent using the ASA postulate?
A. the measures of the missing angles
B. the measures of sides and
C. the measures of sides and
D. the measures of sides and
If you are to use the ASA postulate to prove congruence, you need to have two pairs of congruent angles and the included side, the side in between the pairs of congruent angles. The side in between the two marked angles in is side . The side in between the two marked angles in is side You would need the measures of sides and to prove congruence. The correct answer is C.
AAS Congruence
Another way you can prove congruence between two triangles is using two angles and the non-included side.
Angle-Angle-Side (AAS) Congruence Theorem: If two angles and a non-included side in one triangle are congruent to two corresponding angles and a non-included side in another triangle, then the triangles are congruent.
This is a theorem because it can
be proven. First, we will do an example to see why this theorem is true, then we will prove it formally. Like the ASA postulate, the AAS theorem uses two angles and a side to prove triangle congruence. However, the order of the letters (and the angles and sides they stand for) is different.
The AAS theorem is equivalent to the ASA postulate because when you know the measure of two angles in a triangle, you also know the measure of the third angle. The pair of congruent sides in the triangles will determine the size of the two triangles.
Example 2
What information would you need to prove that these two triangles were congruent using the AAS theorem?
A. the measures of sides and
B. the measures of sides and
C. the measures of and
D. the measures of angles and
If you are to use the AAS theorem to prove congruence, you need to know that pairs of two angles are congruent and the pair of sides adjacent to one of the given angles are congruent. You already have one side and its adjacent angle, but you still need another angle. It needs to be the angle not touching the known side, rather than adjacent to it. Therefore, you need to find the measures of and to prove congruence. The correct answer is D.
When you use AAS (or any triangle congruence postulate) to show that two triangles are congruent, you need to make sure that the corresponding pairs of angles and sides actually align. For instance, look at the diagram below:
Even though two pairs of angles and one pair of sides are congruent in the triangles, these triangles are NOT congruent. Why? Notice that the marked side in is which is between the unmarked angle and the angle with two arcs. However in , the marked side is between the unmarked angle and the angle with one arc. As the corresponding parts do not match up, you cannot use AAS to say these triangles are congruent.
AAS and ASA
The AAS triangle congruence theorem is logically the exact same as the ASA triangle congruence postulate. Look at the following diagrams to see why.
Since and , we can conclude from the third angle theorem that . This is because the sum of the measures of the three angles in each triangle is and if we know the measures of two of the angles, then the measure of the third angle is already determined. Thus, marking , the diagram becomes this:
Now we can see that (A), (S), and (A), which shows that by ASA.
Proving Triangles Congruent
In geometry we use proofs to show something is true. You have seen a few proofs already—they are a special form of argument in which you have to justify every step of the argument with a reason. Valid reasons are definitions, postulates, or results from other proofs.
One way to organize your thoughts when writing a proof is to use a two-column proof. This is probably the most common kind of proof in geometry, and it has a specific format. In the left column you write statements that lead to what you want to prove. In the right hand column, you write a reason for each step you take. Most proofs begin with the “given” information, and the conclusion is the statement you are trying to prove. Here’s an example:
Example 3
Create a two-column proof for the statement below.
Given: is the bisector of , and
Prove:
Remember that each step in a proof must be clearly explained. You should formulate a strategy before you begin the proof. Since you are trying to prove the two triangles congruent, you should look for congruence between the sides and angles. You know that if you can prove SSS, ASA, or AAS, you can prove congruence. Since the given information provides two pairs of congruent angles, you will most likely be able to show the triangles are congruent using the ASA postulate or the AAS theorem. Notice that both triangles share one side. We know that side is congruent to itself , and now you have pairs of two congruent angles and non-included sides. You can use the AAS congruence theorem to prove the triangles are congruent.
Statement Reason
1. is the bisector of
1. Given
2.
2. Definition of an angle bisector (a bisector divides an angle into two congruent angles)
3.
3. Given
4.
4. Reflexive Property
5.
5. AAS Congruence Theorem (if two pairs of angles and the corresponding non-included sides are congruent, then the triangles are congruent)
Notice how the markings in the triangles help in the proof. Whenever you do proofs, use arcs in the angles and tic marks to show congruent angles and sides.
Flow Proofs
Though two-column proofs are the most traditional style (in geometry textbooks, at least!), there are many different ways of solving problems in geometry. We already wrote a paragraph proof in an earlier lesson that simply described, step by step, the rationale behind an assertion (when we showed why AAS is logically equivalent to ASA). The two-column style is easy to read and organizes ideas clearly. Some students, however, prefer flow proofs. Flow proofs show the relationships between ideas more explicitly by using a chart that shows how one idea will lead to the next. Like two-column proofs, it is helpful to always remember the end goal so you can identify what it is you need to prove. Sometimes it is easier to work backwards!
The next example repeats the same proof as the one above, but displayed in a flow style, rather than two columns.
Example 4
Create a flow proof for the statement below.
Given: is the bisector of and
Prove:
As you can see from these two proofs of the theorem, there are many different ways of expressing the same information. It is important that you become familiar with proving things using all of these styles because you may find that different types of proofs are better suited for different theorems.
Lesson Summary
In this lesson, we explored triangle congruence. Specifically, we have learned to:
Understand and apply the ASA Congruence Postulate.
Understand and apply the AAS Congruence Postulate.
Understand and practice Two-Column Proofs.
Understand and practice Flow Proofs.
These skills will help you understand issues of congruence involving triangles. Always look for triangles in diagrams, maps, and other mathematical representations.
Points to Consider
Now that you have been exposed to the SAS and AAS postulates, there are even more triangle congruence postulates to explore. The next lesson deals with SAS and HL proofs.
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, what other congruence statements do you now know based on your answers to 1 and 2?
Use the following diagram 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 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?
Complete the steps of this two-column proof:
Given , and
Prove:
Note: You cannot assume that and are collinear or that and are collinear.
Statement Reason
1. 1. Given
2. 2
. ________
3. ________ 3. Given
4. _______ 4. _______ triangle congruence postulate
5. 5. ________________________________
Bonus question: Why do we have to use three letters to name and , while we can use only one letter to name or ?
Review Answers
AAS triangle congruence postulate
and
No congruence statement is possible
We can’t use either AAS or ASA because the corresponding parts do not match up
. This is still true by the third angle theorem, even if the triangles are not congruent.
ASA triangle congruence postulate
, and
Statement Reason
1.
1. Given
2.
2. Given
3.
3. Given
4.
4. AAS Triangle Congruence Postulate
5.
5. Definition of congruent triangles (if two triangles are then all corresponding parts are also ).
We can use one letter to name an angle when there is no ambiguity. So at point in the diagram for 10 there is only one possible angle. At point there are four angles, so we use the “full name” of the angles to be specific!