Congruence Criteria for Triangles—ASA and SSS
Use the provided 30° angle as one base angle of an isosceles triangle. Use a compass and straight edge to construct an appropriate isosceles triangle around it.
Compare your constructed isosceles triangle with a neighbor’s. Does using a given angle measure guarantee that all the triangles constructed in class have corresponding sides of equal lengths?
Today we are going to examine two more triangle congruence criteria, Angle-Side-Angle (ASA) and Side-Side-Side (SSS), to add to the SAS criteria we have already learned. We begin with the ASA criteria.
ANGLE-SIDE-ANGLE TRIANGLE CONGRUENCE CRITERIA (ASA): Given two triangles △ 𝐴𝐵𝐶 and △ 𝐴′𝐵′𝐶′, if 𝑚∠𝐶𝐴𝐵 = 𝑚∠𝐶 ′𝐴 ′𝐵 ′ (Angle), 𝐴𝐵 = 𝐴′𝐵′ (Side), and 𝑚∠𝐶𝐵𝐴 = 𝑚∠𝐶 ′𝐵 ′𝐴 ′ (Angle), then the triangles are congruent.
We do not begin at the very beginning of this proof. Revisit your notes on the SAS proof, and recall that there are three cases to consider when comparing two triangles. In the most general case, when comparing two distinct triangles, we translate one vertex to another (choose congruent corresponding angles). A rotation brings congruent, corresponding sides together. Since the ASA criteria allows for these steps, we begin here.
In order to map △ 𝐴𝐵𝐶′′′ to △ 𝐴𝐵𝐶, we apply a reflection 𝑟 across the line 𝐴𝐵. A reflection maps 𝐴 to 𝐴 and 𝐵 to 𝐵, since they are on line 𝐴𝐵. However, we say that 𝑟(𝐶′′′) = 𝐶. Though we know that 𝑟(𝐶′′′) is now in the same halfplane of line 𝐴𝐵 as 𝐶, we cannot assume that 𝐶′′′ maps to 𝐶. So we have 𝑟(△ 𝐴𝐵𝐶′′′) = △ 𝐴𝐵𝐶. To prove the theorem, we need to verify that 𝐶∗ is 𝐶.
By hypothesis, we know that ∠𝐶𝐴𝐵 ≅ ∠𝐶′′′𝐴𝐵 (recall that ∠𝐶′′′𝐴𝐵 is the result of two rigid motions of ∠𝐶 ′𝐴 ′𝐵 ′ , so must have the same angle measure as ∠𝐶 ′𝐴 ′𝐵 ′ ). Similarly, ∠𝐶𝐵𝐴 ≅ ∠𝐶′′′𝐵𝐴. Since ∠𝐶𝐴𝐵 ≅ 𝑟(∠𝐶 ′′′𝐴𝐵) ≅ ∠𝐶∗𝐴𝐵, and 𝐶 and 𝐶* are in the same half-plane of line 𝐴𝐵, we conclude that 𝐴𝐶 and 𝐴𝐶∗ must actually be the same ray. Because the points 𝐴 and 𝐶∗ define the same ray as 𝐴𝐶 , the point 𝐶∗ must be a point somewhere on 𝐴𝐶. Using the second equality of angles, ∠𝐶𝐵𝐴 ≅ 𝑟(∠𝐶 ′′′𝐵𝐴) ≅ ∠𝐶 ∗𝐵𝐴, we can also conclude that 𝐵𝐶 and 𝐵𝐶∗ must be the same ray. Therefore, the point 𝐶∗ must also be on 𝐵𝐶 . Since 𝐶∗ is on both 𝐴𝐶 and 𝐵𝐶, and the two rays only have one point in common, namely 𝐶, we conclude that 𝐶 = 𝐶∗.
We have now used a series of rigid motions to map two triangles onto one another that meet the ASA criteria.
SIDE-SIDE-SIDE TRIANGLE CONGRUENCE CRITERIA (SSS): Given two triangles △ 𝐴𝐵𝐶 and △ 𝐴’𝐵’𝐶’, if 𝐴𝐵 = 𝐴′𝐵′ (Side), 𝐴𝐶 = 𝐴′𝐶′ (Side), and 𝐵𝐶 = 𝐵′𝐶′ (Side), then the triangles are congruent.
PROOF: Again, we do not need to start at the beginning of this proof, but assume there is a congruence that brings a pair of corresponding sides together, namely the longest side of each triangle.
Without any information about the angles of the triangles, we cannot perform a reflection as we have in the proofs for SAS and ASA. What can we do? First we add a construction: Draw an auxiliary line from 𝐵 to 𝐵′, and label the angles created by the auxiliary line as 𝑟, 𝑠, 𝑡, and 𝑢.
Since 𝐴𝐵 = 𝐴𝐵′ and 𝐶𝐵 = 𝐶𝐵′, △ 𝐴𝐵𝐵′ and △ 𝐶𝐵𝐵′ are both isosceles triangles respectively by definition. Therefore, 𝑟 = 𝑠 because they are base angles of an isosceles triangle 𝐴𝐵𝐵′. Similarly, 𝑚∠𝑡 = 𝑚∠𝑢 because they are base angles of △ 𝐶𝐵𝐵′. Hence, ∠𝐴𝐵𝐶 = 𝑚∠𝑟 + 𝑚∠𝑡 = 𝑚∠𝑠 + 𝑚∠𝑢 = 𝑚∠𝐴𝐵 ′𝐶. Since 𝑚∠𝐴𝐵𝐶 = 𝑚∠𝐴𝐵′𝐶, we say that △ 𝐴𝐵𝐶 ≅△ 𝐴𝐵′𝐶 by SAS. We have now used a series of rigid motions and a construction to map two triangles that meet the SSS criteria onto one another. Note that when using the Side-Side-Side triangle congruence criteria as a reason in a proof, you need only state the congruence and SSS. Similarly, when using the Angle-Side-Angle congruence criteria in a proof, you need only state the congruence and ASA. Now we have three triangle congruence criteria at our disposal: SAS, ASA, and SSS. We use these criteria to determine whether or not pairs of triangles are congruent.
Based on the information provided, determine whether a congruence exists between triangles. If a congruence exists between triangles or if multiple congruencies exist, state the congruencies and the criteria used to determine them.
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