Introduction to Oblique Triangles
Not every triangle has a right angle. When SOH CAH TOA can't help, you need new tools — the Law of Sines and the Law of Cosines.
Introduction
An oblique triangle is any triangle that does notcontain a 90° angle. Since there's no hypotenuse, you can't use right-triangle ratios directly. Instead, you'll need the Law of Sines (Lessons 7.2–7.3) and the Law of Cosines (7.4–7.5) to find missing sides and angles.
Past Knowledge
Right-triangle trig (SOH CAH TOA), triangle angle-sum theorem (angles add to 180°).
Today's Goal
Classify oblique triangles by given information (AAS, ASA, SSA, SAS, SSS) and determine which law to apply.
Future Success
Correctly identifying the case (AAS vs. SSA vs. SAS) is the first step for every problem in Lessons 7.2–7.7.
Key Concepts
What Is an Oblique Triangle?
A triangle with no right angle. It may be:
- Acute — all three angles less than 90°
- Obtuse — one angle greater than 90°
Standard Labeling Convention
In an oblique triangle :
- Angles: , ,
- Opposite sides: , , (side is opposite angle , etc.)
Which Law Do I Use?
| Given Information | Abbreviation | Use… |
|---|---|---|
| Two angles + one side | AAS or ASA | Law of Sines |
| Two sides + angle opposite one | SSA | Law of Sines (ambiguous case) |
| Two sides + included angle | SAS | Law of Cosines |
| Three sides | SSS | Law of Cosines |
Quick Decision Rule
If you know an angle–side pair (an angle and its opposite side), start with the Law of Sines. Otherwise, use the Law of Cosines.
Worked Examples
Classifying By Given Information
Given: , , . Classify and choose a law.
Step 1: We have two angles and a side → AAS.
Step 2: We know angle and its opposite side → we have a pair.
Answer: AAS case → use the Law of Sines.
Identifying the Ambiguous Case
Given: , , . Classify and choose a law.
Step 1: Two sides and an angle opposite one of them → SSA.
Step 2: SSA is the ambiguous case — there may be 0, 1, or 2 valid triangles.
Answer: SSA case → use the Law of Sines (with caution — see Lesson 7.3).
SAS vs. SSA Distinction
Given: , , . Classify and choose a law.
Step 1: Two sides () and angle — is the angle between the sides?
Step 2: Side is opposite , side is opposite , so angle is between sides and . This is SAS.
Answer: SAS case → use the Law of Cosines.
Common Pitfalls
Trying SOH CAH TOA on a Non-Right Triangle
SOH CAH TOA only works when there's a right angle. For oblique triangles, you must use the Law of Sines or Law of Cosines.
Confusing SAS with SSA
In SAS, the angle is between the two sides. In SSA, the angle is opposite one of them. This distinction determines which law to use and whether the ambiguous case applies.
Real-Life Applications
Surveying & Land Measurement
Surveyors routinely measure property boundaries and terrain features where right angles are rare. By measuring distances and angles with a theodolite, they define oblique triangles and use the Law of Sines or Cosines to calculate inaccessible distances — such as the width of a river or the height of a mountain peak.
Practice Quiz
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