Two triangles each have angles of 45° and 70°, and a corresponding non-included side of 5 cm. Which congruence criterion establishes they are congruent?
AASA, because two angles are known
BAAS, because two angles and a non-included side are congruent
CAAA, because all three angles can be determined
DSSA, because a side and two angle measurements are given
This is AAS: two angles and a side that is not between them. AAA is wrong because it proves only similarity, not congruence — triangles with the same angles can have different sizes. SSA is wrong because it's two sides and a non-included angle, which is not the configuration here. AAS works because the Triangle Angle Sum Theorem gives you the third angle, reducing the situation to ASA.
Question 2 Multiple Choice
AAS is a valid congruence criterion even though the known side is not between the two known angles. Why?
ABecause the non-included side determines scale more reliably than an included side
BBecause the Triangle Angle Sum Theorem gives you the third angle for free, so you effectively have all three angles plus one side — which fully determines the triangle
CBecause any two triangles with two matching angles are automatically congruent
DBecause AAS is actually just another name for ASA when the triangle is obtuse
The key is the angle sum theorem: if you know two angles, you know the third (180° − the other two). Now you have all three angles and one side, which is enough to determine a unique triangle. In effect, AAS secretly reduces to ASA — identify which two angles the known side falls between once you've computed the third angle. The common mistake is thinking the position of the known side makes AAS weaker; it doesn't, because angles constrain direction, and fixing all directions plus any one side locks down scale.
Question 3 True / False
Two triangles with most three pairs of corresponding angles equal are congruent.
TTrue
FFalse
Answer: False
AAA (Angle-Angle-Angle) proves that two triangles are similar — same shape — but not necessarily congruent. A small equilateral triangle and a large equilateral triangle both have three 60° angles but are clearly different sizes. Congruence requires fixing both shape and size; fixing all angles fixes shape but leaves scale free. You need at least one side to pin down the size.
Question 4 True / False
Whenever you have AAS, you can invoke the Triangle Angle Sum Theorem to derive the third angle and then reinterpret the configuration as ASA.
TTrue
FFalse
Answer: True
This is exactly why AAS works. The three angles of any triangle sum to 180°, so knowing two angles determines the third. Once you have all three angles, the known side sits between two specific angles — whichever pair it falls between gives you ASA. This derivation is not just a proof technique; it is the conceptual reason AAS is a valid criterion.
Question 5 Short Answer
Explain why ASA and AAS succeed as congruence criteria but AAA does not, and why SSA also fails.
Think about your answer, then reveal below.
Model answer: ASA and AAS both fix the triangle's shape and scale completely. With two angles determined, the shape is set (all three angles determine a unique shape class, since the third is forced by the angle sum). One side then fixes the scale — there is no room left for the triangle to be larger or smaller. AAA fixes shape but not scale, so infinitely many similar triangles satisfy it. SSA fixes one angle and two sides but leaves ambiguity: depending on the lengths, two different triangles can satisfy the same SSA conditions (the 'ambiguous case'), so it does not guarantee uniqueness.
The deeper principle is that angles constrain direction and shape; sides constrain length and scale. You need enough information to eliminate all degrees of freedom. Two angles eliminate all shape freedom; one side then eliminates scale. But SSA has a side on the 'wrong' side of the fixed angle, which allows the opposite vertex to swing into two positions — the classic ambiguous case.