Oblique Triangles Summary

Pre-Calculus · Axiom Academy

Everything from this unit in one place: the Law of Sines, the ambiguous SSA case, the Law of Cosines, two ways to find area, and how to choose the right tool. An oblique triangle has no right angle, so right-triangle trig (SOH-CAH-TOA) no longer applies — but the Law of Sines and Law of Cosines solve any triangle once you know enough of its parts. Match the law to the given parts: Law of Sines for AAS, ASA, and SSA (an angle paired with its opposite side); Law of Cosines for SAS and SSS (no such angle–side pair to start from). The SSA case is ambiguous : the same given parts can produce zero, one, or two different triangles. The critical height decides which. Area has two go-to formulas: when you know two sides and the included angle (SAS), and Heron's formula when you know all three sides (SSS). Each side is proportional to the sine of the angle across from it . Use it whenever a known angle is paired with its opposite side, plus one more piece — the AAS , ASA , and SSA cases. When to use: you have a complete angle–opposite-side pair. Watch out for: the SSA setup — it may give two triangles (see below). A generalization of the Pythagorean theorem: the term corrects for the angle not being . Use it for SAS (find the third side) and SSS (rearrange to find an angle). Watch out for: there is no opposite angle–side pair, so the Law of Sines can't even start here. Core Concept The Ambiguous Case (SSA)

This is the written version of the interactive lesson above. See the full Pre-Calculus course.