Congruence Criteria for Triangles—SAS
Answer the following question. Then discuss your answer with a partner. Do you think it is possible to know whether there is a rigid motion that takes one triangle to another without actually showing the particular rigid motion? Why or why not?
It is true that we do not need to show the rigid motion to be able to know that there is one. We are going to show that there are criteria that refer to a few parts of the two triangles and a correspondence between them that guarantee congruency (i.e., existence of rigid motion). We start with the Side-Angle-Side (SAS) criteria.
SIDE-ANGLE-SIDE TRIANGLE CONGRUENCE CRITERIA (SAS): Given two triangles △ 𝐴𝐵𝐶 and △ 𝐴′𝐵′𝐶′ so that 𝐴𝐵 = 𝐴′𝐵′ (Side), 𝑚∠𝐴 = 𝑚∠𝐴′ (Angle), and 𝐴𝐶 = 𝐴 ′𝐶 ′ (Side). Then the triangles are congruent.
The steps below show the most general case for determining a congruence between two triangles that satisfy the SAS criteria. Note that not all steps are needed for every pair of triangles. For example, sometimes the triangles already share a vertex. Sometimes a reflection is needed, sometimes not. It is important to understand that we can always use some or all of the steps below to determine a congruence between the two triangles that satisfies the SAS criteria.
PROOF: Provided the two distinct triangles below, assume 𝐴𝐵 = 𝐴′𝐵′ (Side), 𝑚∠𝐴 = 𝑚∠𝐴′ (Angle), and 𝐴𝐶 = 𝐴′𝐶′ (Side).
By our definition of congruence, we have to find a composition of rigid motions that maps △ 𝐴′𝐵′𝐶′ to △ 𝐴𝐵𝐶. We must find a congruence 𝐹 so that 𝐹(△ 𝐴′𝐵′𝐶′)= △ 𝐴𝐵𝐶. First, use a translation 𝑇 to map a common vertex. Which two points determine the appropriate vector? Can any other pair of points be used? ________ Why or why not?
State the vector in the picture below that can be used to translate △ 𝐴′𝐵′𝐶′. _____________ Using a dotted line, draw an intermediate position of △ 𝐴′𝐵′𝐶′ as it moves along the vector:
After the translation (below), 𝑇𝐴′𝐴(△ 𝐴′𝐵′𝐶′) shares one vertex with △ 𝐴𝐵𝐶, 𝐴. In fact, we can say 𝑇𝐴′𝐴(△ _____________ ) = △ _____________. Next, use a clockwise rotation 𝑅∠𝐶𝐴𝐶′′ to bring the side 𝐴𝐶′′ ̅̅̅̅̅ to 𝐴𝐶 (or a counterclockwise rotation to bring 𝐴𝐵′′ ̅̅̅̅̅̅to 𝐴𝐵)
A rotation of appropriate measure maps 𝐴𝐶′′ to 𝐴𝐶 , but how can we be sure that vertex 𝐶′′ maps to 𝐶? Recall that part of our assumption is that the lengths of sides in question are equal, ensuring that the rotation maps 𝐶′′ to 𝐶. (𝐴𝐶 = 𝐴𝐶′′; the translation performed is a rigid motion, and thereby did not alter the length when 𝐴𝐶′ became 𝐴𝐶′′.)
After the rotation 𝑅∠𝐶𝐴𝐶 ′′ (△ 𝐴𝐵′′𝐶′′), a total of two vertices are shared with △ 𝐴𝐵𝐶, 𝐴 and 𝐶.
Finally, if 𝐵′′′ and 𝐵 are on opposite sides of the line that joins 𝐴𝐶, a reflection 𝑟𝐴𝐶 brings 𝐵′′′ to the same side as 𝐵.
Since a reflection is a rigid motion and it preserves angle measures, we know that 𝑚∠𝐵 ′′′𝐴𝐶 = 𝑚∠𝐵𝐴𝐶 and so 𝐴𝐵′′′ maps to 𝐴𝐵 . If, however, 𝐴𝐵′′′ coincides with 𝐴𝐵 , can we be certain that 𝐵′′′ actually maps to 𝐵? We can, because not only are we certain that the rays coincide but also by our assumption that 𝐴𝐵 = 𝐴𝐵′′′. (Our assumption began as 𝐴𝐵 = 𝐴 ′𝐵 ′ , but the translation and rotation have preserved this length now as 𝐴𝐵′′′.) Taken together, these two pieces of information ensure that the reflection over 𝐴𝐶̅̅̅̅ brings 𝐵′′′ to 𝐵.
Another way to visually confirm this is to draw the marks of the __________ construction for 𝐴𝐶
Write the transformations used to correctly notate the congruence (the composition of transformations) that take △ 𝐴 ′𝐵 ′𝐶 ′ ≅ △ 𝐴𝐵𝐶:
We have now shown a sequence of rigid motions that takes △ 𝐴′𝐵′𝐶′ to △ 𝐴𝐵𝐶 with the use of just three criteria from each triangle: two sides and an included angle. Given any two distinct triangles, we could perform a similar proof. There is another situation when the triangles are not distinct, where a modified proof is needed to show that the triangles map onto each other. Examine these below. Note that when using the Side-Angle-Side triangle congruence criteria as a reason in a proof, you need only state the congruence and SAS.
What if we had the SAS criteria for two triangles that were not distinct? Consider the following two cases. How would the transformations needed to demonstrate congruence change?
Justify whether the triangles meet the SAS congruence criteria; explicitly state which pairs of sides or angles are congruent and why. If the triangles do meet the SAS congruence criteria, describe the rigid motion(s) that would map one triangle onto the other.
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