I’m getting ready for a geometry test and I keep second-guessing myself on congruent triangles, especially SAS vs SSA.
Here’s the setup: I’ve got two triangles, ABC and DEF. In the picture, AB = DE = 5, AC = DF = 7, and ∠A = ∠D = 40°. The catch is that the triangles look like mirror images-like the 40° angle opens in opposite directions in each drawing.
My gut says they’re congruent by SAS because the two sides and the angle between them match. But then I start worrying: does the mirror-image thing matter? Am I actually using the angle that’s between the two sides I matched, or did I accidentally fall into the SSA trap without realizing it? (My brain keeps yelling “SAS!” then “No, SSA!” and now I’m confused.)
Follow-up question: when I write something like ΔABC ≅ ΔDEF, how do I pick the order so that the corresponding vertices line up correctly? Is there a quick way to check that I’m matching the included angle to the right pair of sides? And bonus: does SSA ever work (like in right triangles), or should I just avoid it unless there’s something special given?
Sorry if I’m overthinking this-just trying not to miss an easy point on the test!
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One Response
Your gut’s right-this is SAS: think of two sticks joined by a fixed hinge angle; AB=5 and AC=7 with ∠A=40° at their shared endpoint locks the triangle (a mirror flip is still congruent), and to order it write ΔABC ≅ ΔDEF by lining up the equal-angle vertex A↔D, then AB↔DE gives B↔E and AC↔DF gives C↔F. Example: BC = √(5²+7²−2·5·7·cos40°) ≈ 4.51 (unique), whereas if the 40° were opposite the 5 (SSA) you’d get two triangles since 7·sin40° ≈ 4.50 < 5 < 7; SSA only proves congruence in the right-triangle HL case-see https://www.khanacademy.org/math/geometry/hs-geo-congruence/hs-geo-triangle-congruence/a/congruent-triangles.