4.8: Proofs with CPCTC
Learning Objectives
- Apply various triangle congruence postulates and theorems.
- Know the ways in which you can prove parts of a triangle congruent.
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 ways that can prove congruence, and it is important to know which combinations of sides and angles do not prove congruence.
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 | Picture | 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 | The hypotenuse and a leg in a right triangle | Yes |
Example 1
What rule can prove that the triangles below are congruent?
A. SSS
B. SAS
C. ASA
D. AAS
The two triangles in the picture have two pairs of congruent angles (\begin{align*}\angle DCE \cong \angle ORN\end{align*} and \begin{align*}\angle CED \cong \angle RNO\end{align*}) and one pair of corresponding congruent sides \begin{align*}(\overline{DC} \cong \overline{OR})\end{align*}.
So, the triangle congruence postulate you choose must have two A’s (for the angles) and one \begin{align*}S\end{align*} (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.
Reading Check:
1. The two triangles below are congruent because of the SAS postulate. Mark the SAS congruent parts with tic marks and arcs:
2. True/False: SSS, SAS, ASA, AAS, and HL (in a right triangle) are 5 different ways to prove that triangles are congruent.
\begin{align*}{\;}\end{align*}
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. This may take a bit of thought, and you must use some deductive reasoning (finding conclusions based on facts) to find the missing parts.
When you were creating 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.
Recall that, if two triangles are congruent, then all pairs of corresponding sides and all pairs of corresponding angles are congruent.
We say that the Corresponding Parts (sides and angles) of Congruent Triangles are Congruent, or CPCTC.
Reading Check:
1. What is a part of a triangle?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. What does it mean for parts to be corresponding?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
3. Explain in your own words what CPCTC means.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
How do we use the concept of CPCTC in a proof?
Check out the example on the following page…
Example 2
How could you prove that segment \begin{align*}AB\end{align*} is congruent to segment \begin{align*}DE\end{align*} in the diagram below?
We can see that \begin{align*}\overline{BC} \cong \overline{CE}\end{align*} and \begin{align*}\overline{AC} \cong \overline{CD}\end{align*} because of the tic marks in the figure.
Think of all of the postulates and theorems (in the chart at the beginning of Lesson 7 and in your graphic organizer) that have 2 S’s (SSS and SAS). We may be able to use SSS or SAS to show the triangles are congruent. However, to use SSS, we would need \begin{align*}\overline{AB} \cong \overline{DE}\end{align*} and we cannot yet make this assumption.
Can we show that two of the angles are congruent?
Notice that \begin{align*}\angle BCA\end{align*} and \begin{align*}\angle ECD\end{align*} are vertical angles (non-adjacent 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 \begin{align*}\angle BCA \cong \angle ECD\end{align*}.
We now have two congruent sides and a congruent angle between the sides!
- The congruent sides are:
___________ \begin{align*}\cong\end{align*} ___________ and ___________ \begin{align*}\cong\end{align*} ___________
- The congruent angles are:
_______________ \begin{align*}\cong\end{align*} _______________
By putting all of this information together, you can confirm that \begin{align*}\Delta ABC \cong \Delta DEC\end{align*} by the SAS Postulate.
Finally, if the two triangles are congruent, then their corresponding parts are congruent.
Therefore segment \begin{align*}AB\end{align*} is congruent to segment \begin{align*}DE\end{align*} by CPCTC.
Reading Check:
1. In the space below, draw a picture of two triangles that are congruent because of the ASA postulate. Make sure to mark the congruent parts with tic marks and arcs!
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. If you know that 2 pairs of angles and 1 pair of sides are congruent in your picture above, what other parts of the triangles are congruent? (Hint: there are 3 answers!)
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
3. Your answers to question #2 above are true because of CPCTC. The letters CPCTC stand for: (Fill in the blanks) ______________________ ______________________ of ________________________ ________________________ are _________________________ .
Finding Distances (a real-word application)
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 is exactly halfway between two pairs of cities: it is halfway between Camden and Grenata AND it is halfway between 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 triangles drawn on the map are congruent.
Since you know that the distance from Camden to Meridian is the same as Meridian to Grenata (since Meridian is halfway between those cities), those two sides of the triangles are congruent. You can make a tic mark on these sides.
Similarly, since the distance from Lowell to Meridian is the same as Meridian to Morsetown (again because Meridian is halfway between), those two sides are also a congruent pair and you can give them congruent tic marks.
Finally, you can tell that the angles between these lines (at the intersection where Meridian is) are also congruent because they are vertical angles.
With all of your congruent sides and angles marked, your map will look like this:
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 5 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.
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