Build two different triangles from the same side-side-angle measurements, and see why SSA can't be trusted.
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Two sides and an angle work only when the angle is between them.
Two sides and an angle are enough to guarantee congruence when the angle lies BETWEEN the two sides, which is the SAS condition. The same information with the angle somewhere else is SSA, and it can be satisfied by two different triangles. There is one important special case where SSA does work, however, and it comes up when the triangle has a right angle.
The same two sides, but the angle is no longer between them.
Consider two triangles, and , where , , and . Are they exactly congruent?
Building the triangle with PQ = 6 cm, 30 degrees at P and a 4 cm arc.
Problem. Construct a triangle with base , an angle at of , and the side opposite that base region measuring from .
The arc from Q cuts the ray from P at both R and S.
The arc from Q crosses the ray in two places, revealing two possible triangles.
Triangle PQR and triangle PQS side by side with identical measurements.
Same three measurements, completely different triangles.
Arrange the construction steps that expose the two triangles.
Order the steps to construct the SSA counterexample.
Sort measurement sets into the reliable case and the unreliable one.
In two triangles, you are given , , and . Which congruence condition does this satisfy?