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Congruence, Proof, and Constructions

Congruent Triangles

According to the definition of congruence, two figures are congruent if there is a rigid motion that maps one onto the other. In comparison, for polygons, there is another method that makes it possible to determine congruence without involving rigid motions. That method will be developed and understood in this lesson.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Determining If Two Triangles Are Congruent

By applying rigid motions, investigate whether the given pair of triangles are congruent or not.

  • The left-hand side triangle can be translated by dragging it.
  • The left-hand side triangle can be rotated about the movable point by clicking and dragging any of its vertices.
  • The left-hand side triangle can be reflected across the line shown, which can be set by moving its points.
Comparing different pairs of triangles
When the triangles are not congruent, how are they different? When the triangles are congruent, what do they have in common?

Discussion

Accounting For Corresponding Parts

When a rigid motion is applied to a polygon, the resulting image can be seen as the polygon formed by the image of each part of the original polygon under the same rigid motion.
Mapping each part of one polygon to its corresponing part
There is a correspondence between the parts of the preimage and the parts of the image. This relationship leads to the following definition, which will simplify future discussions.

Concept

Corresponding Parts

Consider two figures. One figure is the image of the other under a transformation. The pairs formed by a part of the preimage — a side, angle, or vertex — and the image of that part are called corresponding parts. For example, in the following applet, and are corresponding vertices.
Corresponding Parts of Two Triangles
By clicking on each part of the corresponding part on will be highlighted. Note that the figures and their corresponding parts can either be congruent or similar.

Discussion

Properties of Congruent Triangles

When two triangles are congruent, it seems natural to think that the corresponding parts are congruent as well. What about the same concept when the statement is reversed? That is to say, if the corresponding parts of two triangles are congruent, are the triangles congruent? The following claim answers this question.

Rule

Congruent Triangles

Two triangles are congruent if and only if their corresponding sides and angles are congruent.

Triangles ABC and DEF

Using the triangles shown, this claim can be written algebraically as follows.

Proof

This proof will be developed based on the given diagram, but it is valid for any pair of triangles. The proof of this biconditional statement consists of two parts, one for each direction.

  1. If and are congruent, then their corresponding sides and angles are congruent.
  2. If the corresponding sides and angles of and are congruent, then the triangles are congruent.

Part

By definition of congruent figures, if the triangles are congruent there is a rigid motion or sequence of rigid motions that maps onto
Mapping ABC onto DEF
Because rigid motions preserve side lengths, and its image have the same length, that is, Therefore, Similarly for the other two side lengths. Furthermore, rigid motions preserve angle measures. Then, and its image have the same measure, that is, Therefore, Similarly for the remaining angles. That way, it has been shown that if two triangles are congruent, then their corresponding sides and angles are congruent.

Part

To begin, mark the congruent parts on the given diagram.

Marking the congruent parts

The primary purpose is finding a rigid motion or sequence of rigid motions that maps one triangle onto the other. This can be done in several ways, here it is shown one of them.

1

Translate so that one pair of corresponding vertices match
Apply a translation to that maps to If this translation maps onto the proof will be complete.
Translating Triangle ABC
As seen, did not match Therefore, a second rigid motion is needed.

2

Rotate so that one pair of corresponding sides match
Apply a clockwise rotation to about through If the image matches the proof will be complete. Notice this rotation maps onto and therefore, onto
Rotating Triangle DB'C'
As before, the image did not match Thus, a third rigid motion is required.

3

Reflect so that the corresponding sides match
Apply a reflection to across Because reflections preserve angles, is mapped onto and is mapped onto Then, the intersection of the original rays is mapped to the intersection of the image rays
Reflecting Triangle DEC''
This time the image matched

Consequently, through applying different rigid motions, was mapped onto This implies that and are congruent. Then, the proof is complete.

Example

Investigating If Two Triangles Are Congruent

The local chess team in Jordan's town has designed a fancy new logo.
Logo formed by two triangles
Jordan thinks the logo is made of two congruent triangles. Is she correct?
What is the sum of the perimeters of both triangles?

Hint

Use the measuring tool to find the side lengths and angle measures of each triangle. Then, compare them. Are all corresponding parts congruent? If the triangles are congruent, they have equal perimeters.

Solution

To determine whether the logo is made of two congruent triangles, the measures of both triangles need to be found. For simplicity, start by separating the two triangles involved.
Separating the Triangles
Then, find and label the side lengths and angle measures of each triangle.
Labeling the measures of the triangles

As seen, both triangles have the same side lengths and angle measures. That is, all their corresponding sides and angles are congruent. Therefore, the two triangles are congruent. Jordan was correct!

The new logo is made of two congruent triangles.

Because the triangles are congruent, they have the same perimeter. Then, it is enough to find the perimeter of only one of them and then multiply it by Remember, the perimeter is the sum of the side lengths. Finally, the sum of the perimeters of both triangles is twice That can be expressed as

Example

Using Properties of Triangles to Determine Congruence

To show that two triangles are not congruent, it is enough to find only one side or angle of one of the triangles that is not congruent to any side or angle of the second triangle.

Mark, Magdalena, and Kevin each bought a notebook with a triangle design on the cover.

Three books with a triangle in the cover

Are there any pairs of congruent triangles? If so, which ones are congruent?

Answer

No, there are no pairs of congruent triangles.

Hint

To conclude that two triangles are congruent, check that all corresponding sides and angles are congruent.

Solution

Beginning with Mark's notebook, notice that the triangle has a angle, while neither of the other two triangles have a angle. That is, there is a part in Mark's triangle that is not congruent to any part of the other two triangles.

The triangle in Mark's notebook is not congruent to any of the other triangles.

Now, moving on to Madaelena's notebeook. Notice that the triangle has the same angle measures as the triangle in Kevin's notebook.

Magdalena and Kevin's triangle

Before concluding that these two triangles are congruent, remember, all the corresponding parts have to be congruent. Therefore, the sides opposite to the angle of each triangle should be congruent. Well, that is not the case.

Magdalena and Kevin's triangle

It was a close call, but the triangle in Magdalena's notebook is not congruent to the triangle in Kevin's notebook. As a result, none of the triangles are congruent to the others.

Pop Quiz

Practice Identifying Congruent Triangles

Determine whether the given pair of triangles are congruent.

Random triangles

Example

Investigating Congruent Triangles

The following figure is made of triangles.

Figure formed by 5 triangles with some common sides
How many pairs of congruent triangles are there?
Write the corresponding congruence statements.

Answer

There are four pairs of congruent triangles in the given figure.

Hint

Notice that is a common side for and Remember, if two sides or angles have the same number of marks, they are congruent.

Solution

To determine which pairs of triangles are congruent, pick one of them and compare its parts to the parts of the rest of the triangles. Start by comparing to These two triangles have as a common side.

Triangles AFG and ABF

By applying the concept of reading the marks to check for congruency, it can be said that is not congruent to any side of Consequently, is not congruent to By the same reasoning, is not congruent to either. Next, compare to

Triangles AFG and DEF

Since and have both a measure of they are congruent angles. Furthermore, the rest of the corresponding parts of these two triangles are congruent. Consequently, and are congruent. Similarly, it can be checked that is also congruent to By the Transitive Property of Congruence, a third pair of congruent triangles can be obtained.

Triangles GEF and BDC are congruent by the Transitive Prop. of Congruence

Next, compare to Since and have the same measure, they are congruent. Additionally, is a common side for both triangles and, by the Reflexive Property of Congruence, In addition, the rest of corresponding parts of these two triangles are congruent. Consequently, and are congruent. So far, there are four pairs of congruent triangles. Taking into account the Transitive Property of Congruence, and the fact that is not congruent to it can be said that there are no more pairs of congruent triangles.

Two set of congruent triangles that are not congruent. The sets are not congruent to each other.

Closure

Writing a Congruence Statement

As with polygons, when it comes to writing a triangle congruence statement, the order in which the vertices are written is critical. Naming them in an incorrect order leads to erroneous conclusions. Consider, for example, the following congruent triangles.

Triangles ABC and MQT
Although the triangles are congruent, the congruence statement is incorrect. Why? Because it would lead to the following conclusions about the sides and angles.
Conclusions based on the congruence statement ABC cong MQT
Consequently, always be sure to list the corresponding vertices in the correct order. Furthermore, another important concept to consider is that the claim which helps to determine whether two triangles are congruent is also valid for polygons. In fact, the claim is identical, except that triangles has been replaced by polygons.

Congruent Polygons

Two polygons are congruent if and only if their corresponding sides and angles are congruent.

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