4.7: Triangle Congruence Proofs
Learning Objectives
- Explore properties of triangle congruence.
- Understand and practice flow proofs.
- Understand and practice two-column proofs.
Congruence Properties
In earlier mathematics courses, you have learned concepts like the commutative or associative properties. These concepts help you solve many types of mathematics problems. There are a few properties relating to congruence that will help you solve geometry problems as well. These are especially useful in two-column proofs, which you will learn later in this lesson!
The Reflexive Property of Congruence
The reflexive property of congruence states that any shape is congruent to itself. This may seem obvious, but in a geometric proof, you need to identify every possibility to help you solve a problem. If two triangles share a line segment, you can prove congruence by the reflexive property.
The Reflexive Property says that any shape is _________________________________ to itself.
In the diagram above, you can say that the shared side of the triangles \begin{align*}(\overline{AB})\end{align*}
One way to remember the Reflexive Property is that the word “reflexive” has the same root as “reflection.”
“Reflection” should make you think of a mirror.
When you look in the mirror, you see yourself!
Likewise, the reflexive property says that something is equal to itself.
The Symmetric Property of Congruence
The symmetric property of congruence states that congruence works frontwards and backwards, or in symbols, if \begin{align*}\angle ABC \cong \angle DEF\end{align*}
The Symmetric Property says that frontwards congruence and __________________________ congruence are the same.
One way to remember the symmetric property is that the word “symmetric” means “the same front to back.”
The picture of the shape below is “symmetric” because the left side is the same as the right side (but backwards) and the top is the same as the bottom (but backwards):
The dotted lines down the middle of the shapes can act like a folding line: if you fold the shape over the line, the two sides will be on top of each other. Each half is the same, but backwards, of its other half.
The Transitive Property of Congruence
The transitive property of congruence states that if two shapes are congruent to a third, they are also congruent to each other.
In other words, if \begin{align*}\Delta ABC \cong \Delta JLM\end{align*}
This property is very important in identifying congruence between different shapes.
The Transitive Property says that if a first shape is congruent to a second, and if the second is congruent to a third, then the _________________ shape is congruent to the ________________.
One way to remember the transitive property is that the word “transitive” is similar to the word “train.”
A train has cars that connect to each other. In the transitive property, the equal (or congruent) sign acts like the connecting piece between the cars:
If \begin{align*}a = b\end{align*}, and \begin{align*}b = c\end{align*}, then \begin{align*}a = c\end{align*}.
On a train, this means that the first car (a) is connected to the second car (b), and the second car (b) is connected to the last car (c). So, the first car (a) is connected to the last car (c)!
Graphic Organizer for Lesson 6
Name of Property | Give an example: Write a statement of congruence using this property | In your own words, how can you recognize this property? | What is a good way to remember this property? |
---|---|---|---|
Reflexive | |||
Symmetric | |||
Transitive |
Reading Check:
Name the property used in the following geometric statements:
1. \begin{align*}\angle MLK \cong \angle KLM\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. If \begin{align*}\angle PQR \cong \angle BCD\end{align*} and \begin{align*}\angle BCD \cong \angle XYZ\end{align*}, then \begin{align*}\angle PQR \cong \angle XYZ\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
3. \begin{align*}\angle FGH \cong \angle FGH\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
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, properties, postulates, theorems or results from other proofs.
Proofs show that something is _____________________.
In a proof, you must provide a _______________________________ for each step.
Proofs use deductive reasoning. You will begin with statements that are accepted as facts, which will lead to other statements based on these facts. Each statement and its corresponding reason will be a step in your proof.
Proofs use __________________________________ reasoning.
Do you remember deductive reasoning from Unit 2?
Deductive reasoning is also known as logic.
Deductive reasoning starts with accepted facts or statements which we know are true. Then, we draw conclusions based on those facts.
Why do you think deductive reasoning is the best method for proofs?
Graphic Organizer: List of Properties, Postulates, and Definitions
List of Properties, Postulates, and Definitions that often come up in proofs:
Name of Property, Postulate, etc. | Give an example or explain in your own words |
---|---|
Vertical Angles Theorem | |
Corresponding Angles Postulate | |
Definition of Segment Bisector | |
Definition of Angle Bisector | |
Definition of Perpendicular Bisector | |
Transitive Property |
Flow Proofs
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 (which you will learn next) is generally most popular, 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 example below is displayed in a flow style (rather than the two-column proof you will see next) and provides a good way to organize your information, where one idea flows into the next.
Example 1
Create a flow proof for the statement below.
Given: \begin{align*}\overline{NQ}\end{align*} is the bisector of \begin{align*}\angle MNP\end{align*} and \begin{align*}\angle NMQ \cong \angle NPQ\end{align*}
Prove: \begin{align*}\Delta MNQ \cong \Delta PNQ\end{align*}
You can see the given information on the left side of the flow proof, and the arrows lead to each additional statement based on the given fact. The reason for each statement is written below the statement. The final statement that you are trying to prove is on the right side of the proof.
Two-Column 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 two columns with 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 reasons for each step you take.
- Most proofs begin with the “given” information, and
- The conclusion is the statement you are trying to prove.
A two-column proof often looks like this:
Statements | Reasons |
---|---|
1. Put the “given” statement here. The “given” statement is always first. | 1. Given (always first!) |
2. List each step that you need to create your proof. Each separate statement in your proof gets its own line. | 2. Each statement has a corresponding reason to explain why you are allowed to make the statement. Each reason gets its own line as well. |
3. More statements... | 3. More reasons that justify your statements... |
4. (You may need more...) | 4. |
5. Put the final “prove” statement here. | 5. Put the corresponding reason here. |
A two-column proof has _________________________________ in the left column.
In the right column of a two-column proof, list _________________________________ that correspond to each statement.
The most common proofs in geometry are _____________________________________ proofs.
The example on the following page repeats the same proof as Example 1, but it is in the more traditional two-column style instead of the flow style.
Example 2
Create a two-column proof for the statement below.
Given: \begin{align*}\overline{NQ}\end{align*} is the bisector of \begin{align*}\angle MNP\end{align*} and \begin{align*}\angle NMQ \cong \angle NPQ\end{align*}
Prove: \begin{align*}\Delta MNQ \cong \Delta PNQ \end{align*}
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, SAS, HL, 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 every side is congruent to itself \begin{align*}(\overline{NQ} \cong \overline{NQ})\end{align*}, 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.
Remember, geometric statements go on the left and reasons go on the right. We always start each column with the given information.
Statement | Reason |
---|---|
1. \begin{align*}\overline{NQ}\end{align*} is the bisector of \begin{align*}\angle MNP\end{align*} | 1. Given |
2. \begin{align*}\angle MNQ \cong \angle PNQ\end{align*} | 2. Definition of an angle bisector (a bisector divides an angle into two congruent angles) |
3. \begin{align*}\angle NMQ \cong \angle NPQ\end{align*} | 3. Given |
4. \begin{align*}\overline{NQ} \cong \overline{NQ}\end{align*} | 4. Reflexive Property |
5. \begin{align*}\Delta MNQ \cong\Delta PNQ\end{align*} | 5. AAS Congruence Theorem (if two pairs of angles and the corresponding non-included sides are congruent, then the triangles are congruent) |
Q.E.D.
Q.E.D. is an acronym of the Latin phrase quod erat demonstrandum, which means “that which was to be demonstrated.”
The phrase is traditionally placed in its abbreviated form at the end of a mathematical proof signaling the completion of the proof.
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.
As you can see from the two different styles of 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.
Reading Check:
1. Fill in the blanks:
____________ go on the left side of a two-column proof and ____________ go on the right side.
2. Fill in the blank:
The ____________ statement always goes first in a two-column proof.
3. What type of reasoning are proofs?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
4. Which style of proofs do you like better, flow proofs or two-column proofs? Why?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
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