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LC Math 2 Adv – Additional Triangle Congruence Criteria (LT 8)
Exploratory Challenge
Today we are going to examine four possible triangle congruence criteria, Angle-Angle-Side (AAS), Side-Side-Angle (SSA), AngleAngle-Angle (AAA), and Hypotenuse-Leg (HL). Ultimately, only some of these criteria ensure congruence.
ANGLE-ANGLE-SIDE TRIANGLE CONGRUENCE CRITERIA (AAS): Given two triangles △ 𝐴𝐵𝐶 and △ 𝐴′𝐵′𝐶′. If 𝐴𝐵 = 𝐴′𝐵′ (Side),
𝑚∠𝐵 = 𝑚∠𝐵′ (Angle), and 𝑚∠𝐶 = 𝑚∠𝐶′ (Angle), then the triangles are congruent.
Consider a pair of triangles that meet the AAS criteria. If you knew that two angles of one triangle corresponded to and were equal
in measure to two angles of the other triangle, what conclusions can you draw about the third angle of each triangle?
Given this conclusion, which formerly learned triangle congruence criteria can we use to determine if the pair of triangles are
congruent?
Therefore, the AAS criterion is actually an extension of the
triangle congruence criterion.
HYPOTENUSE-LEG TRIANGLE CONGRUENCE CRITERIA (HL): Given two right triangles △ 𝐴𝐵𝐶 and △ 𝐴′𝐵′𝐶′with right angles 𝐵
and 𝐵′. If 𝐴𝐵 = 𝐴′𝐵′ (Leg) and 𝐴𝐶 = 𝐴′𝐶′ (Hypotenuse), then the triangles are congruent.
As with some of our other proofs, we do not start at the very beginning, but imagine that a congruence exists so that triangles have
been brought together such that 𝐴 = 𝐴′ and 𝐶 = 𝐶′; the hypotenuse acts as a common side to the transformed triangles.
If you knew that two sides of one right triangle corresponded to and were equal in measure to
two angles of the other right triangle, what conclusions can you draw about the third side of
each triangle?
Given this conclusion, which formerly learned triangle congruence criteria can we use to determine if the pair of triangles are
congruent?
Therefore, the AAS criterion is actually an extension of the _____
triangle congruence criterion.
SIDE-SIDE-ANGLE (SSA): Given two triangles △ 𝐴𝐵𝐶 and △ 𝐴′𝐵′𝐶′. If 𝑚∠𝐴 = 𝑚∠𝐴′ (Angle), 𝐴𝐵 = 𝐴′𝐵′ (Side), 𝐵𝐶 = 𝐵′𝐶′
(Side), then the triangles are congruent.
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Construct a triangle using the sides mentioned in the congruence criteria above. Then determine if all triangles constructed with the
congruent corresponding parts will be congruent. Summarize your findings below:
Therefore, the SSA criteria is…
ANGLE-ANGLE-ANGLE (AAA): Given two triangles △ 𝐴𝐵𝐶 and △ 𝐴′𝐵′𝐶′. If 𝑚∠𝐴 = 𝑚∠𝐴′ (Angle), 𝑚∠𝐵 = 𝑚∠𝐵′ (Angle),
𝑚∠𝐶 = 𝑚∠𝐶′ (Angle), then the triangles are congruent.
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Construct a triangle using the sides mentioned in the congruence criteria above. Then determine if all triangles constructed with the
congruent corresponding parts will be congruent. Summarize your findings below:
Therefore, the AAA criteria is…
List all the triangle congruence criteria here:
List the criteria that do not determine congruence here: