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Using Congruent Triangles
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Printed: July 10, 2012
AUTHORS
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C ONCEPT
1
1
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.
TABLE 1.1:
Name
SSS
SAS
ASA
AAS
HL
AAA
SSA
Corresponding congruent parts
Three sides
Two sides and the angle between
them
Two angles and the side between
them
Two angles and a side not between
them
A hypotenuse and a leg in a right
triangle
Three angles
Two sides and an angle not between
them
Does it prove congruence?
Yes
Yes
Yes
Yes
Yes
No—it will create a similar triangle,
but not of the same size
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
Concept 1. Using Congruent Triangles
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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 A0 s (for the angles) and one S (for the side). You
can eliminate choices A and B for this reason. Now that you are deciding between choices C and D, 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.
distance =
q
(x2 − x1 )2 + (y2 − y1 )2
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 4ABC ∼
= 4DECin the diagram below?
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We can already see that BC ∼
= CE and AC ∼
= CD. We may be able to use SSS or SAS to show the triangles are
congruent. However, to use SSS, we would need AB ∼
= DE and there is no obvious way to prove this. Can we show
that two of the angles are congruent? Notice that 6 BCA and 6 ECD are vertical angles (nonadjacent angles made by
the intersection of two lines—i.e., angles on the opposite sides of the intersection).
∼ 6 ECD.
The Vertical Angle Theorem states that all vertical angles are also congruent. So, this tells us that 6 BCA =
∼
Finally, by putting all the information together, you can confirm that 4ABC = 4DEC 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 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.
Concept 1. Using Congruent Triangles
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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
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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.
Concept 1. Using Congruent Triangles
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Now draw a line through the two points of intersections. This forms the perpendicular bisector.
Draw segments connecting the points on the bisector to the original endpoints.
Knowing that the center point is the midpoint of both line segments and that all angles formed around point M are
right angles, you can prove that all four triangles created are congruent by the SAS rule.
Lesson Summary
In this lesson, we explored applications triangle congruence. Specifically, we have learned to:
• Identify various triangles congruence postulates and theorems.
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• Use the fact that corresponding parts of congruent triangles are congruent.
• Find distances using congruent triangles.
• Use construction techniques to create congruent triangles.
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
You now know all the different ways in which you can prove two triangles congruent. In the next chapter you’ll
learn more about isosceles and equilateral triangles.
Review Questions
Use the following diagram for exercises 1-5
1.
2.
3.
4.
5.
Find AB in the diagram above.
Find BC in the diagram above.
What is m6 ABC? How do you know?
What postulate can you use to show 4ABC ∼
= 4JKL?
Use the distance formula to find AC. How can use this to find JL?
6-8: For each pair of triangles, complete the triangle congruence statement, or write “no congruence statement
possible.” Name the triangle congruence postulate you use, or write a sentence to explain why you can’t write a
triangle congruence statement.
6. 4PAL ∼
= 4 __________.
Concept 1. Using Congruent Triangles
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7. 4BIN ∼
= 4__________.
8. 4BOW ∼
= 4__________.
9. In the following diagram, Midtown is exactly halfway between Uptown and Downtown. What is the distance
between Downtown and Lower East Side? How do you know? Write a few sentences to convince a reader
your answer is correct.
10. Given: R is the midpoint of PN and PA||LN Prove: PA ∼
= LN
Review Answers
1. AB = 4
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2. BC = 5
3. 90◦ . Since AB is horizontal (parallel to the x−axis) and BC is vertical (parallel to the y−axis), we can conclude
that they intersect at a right angle.
4. SAS (other
q answers are possible)
√
√
5. AC = (−1 − (−5))2 + (−2 − 3)2 = 16 + 25 = 41. Since the triangles are congruent, we can conclude
√
JL = 41
6. 4PAL ∼
= 4BUD. SAS triangle congruence postulate
7. 4BIN ∼
= 4RAT . SSS triangle congruence postulate
8. No congruence statement is possible; we don’t have enough information.
9. 1.5 km. Since Midtown is the midpoint of the line connecting Uptown and Downtown, we can use the vertical
angle theorem for the angles made by the two lines that meet at Midtown, and then we can conclude that
the triangles are congruent using AAS. If the triangles are congruent then all corresponding parts are also
congruent
TABLE 1.2:
Statement
1. R is the midpoint of PN
2. PR ∼
= RN
3. PAkLN
4. 6 RNL ∼
= 6 RPA
∼
6
5. PAR = 6 NLR
6. 4PAR ∼
= 4NLR
7. PA ∼
LN
=
Reason
1. Given
2. Definition of midpoint
3. Given
4. Alternate interior angles theorem
5. Alternate interior angles theorem
6. AAS triangle congruence postulate
7. Definition of congruent triangles (corresponding
parts are congruent)
Concept 1. Using Congruent Triangles
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