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.
Using the triangles shown, this claim can be written algebraically as follows.
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.
To begin, mark the congruent parts on the given diagram.
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.
Consequently, through applying different rigid motions, was mapped onto This implies that and are congruent. Then, the proof is complete.
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.
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
Mark, Magdalena, and Kevin each bought a notebook with a triangle design on the cover.
Are there any pairs of congruent triangles? If so, which ones are congruent?
No, there are no pairs of congruent triangles.
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.
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.
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.
The following figure is made of triangles.
There are four pairs of congruent triangles in the given figure.
Notice that is a common side for and Remember, if two sides or angles have the same number of marks, they are congruent.
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.
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
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.
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.
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.