Summary: Conditions for Unique, Multiple, or No Triangles
This Grade 7 math skill from Reveal Math, Accelerated summarizes the conditions under which three given measurements produce a unique triangle, multiple triangles, or no triangle at all. Students learn to apply triangle inequality rules and angle-side conditions to predict triangle outcomes from given side lengths and angles.
Key Concepts
When given three side lengths or three angle measures, you can determine how many triangles can be drawn: No Triangle: The given side lengths do not satisfy the Triangle Inequality Theorem (the sum of the two shorter sides is not greater than the longest side), or the given angle measures do not sum to $180^\circ$. Exactly One Unique Triangle: Three valid side lengths are given (Side Side Side). Multiple Triangles: Three valid angle measures are given (Angle Angle Angle).
Common Questions
When do three measurements define a unique triangle?
Three side lengths define a unique triangle if they satisfy the triangle inequality (each side must be less than the sum of the other two). Three angles plus a side also typically define a unique triangle.
When can three measurements produce multiple triangles?
Given two sides and a non-included angle (SSA), it is possible to construct two different valid triangles, making the solution ambiguous.
When is it impossible to form a triangle from three measurements?
If the sum of the two shorter sides is less than or equal to the longest side, no triangle can be formed. This violates the triangle inequality theorem.
What is the triangle inequality theorem?
The triangle inequality states that the sum of any two sides of a triangle must be greater than the third side. If this fails, no triangle exists.
Where is this topic covered in Reveal Math Accelerated?
Conditions for unique, multiple, or no triangles is summarized in the Grade 7 Reveal Math, Accelerated textbook.