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Math101learn.math101.caTriangle Classification
Triangles are classified independently by side lengths and angle measures. By sides: scalene has no equal sides, isosceles at least two, equilateral three.
Classification selects properties and theorems, supports construction, and links side data with angle behaviour.
Intuition and core definition
Triangles are classified independently by side lengths and angle measures. By sides: scalene has no equal sides, isosceles at least two, equilateral three. By angles: acute has three acute angles, right has one right angle, and obtuse has one obtuse angle.
Notation, language, and conditions
Matching tick marks denote congruent sides and arc marks congruent angles. An equilateral triangle is also equiangular with each angle $60^\circ$, hence acute and isosceles under the inclusive “at least two” definition. Side lengths must satisfy the strict triangle inequality: the sum of any two exceeds the third.
Why this idea matters
Triangles can be classified independently by side equality and angle size, so one triangle carries one label from each system.
A dependable method
- Verify three lengths can form a triangle.
- Compare side lengths or tick marks for the side classification.
- Use given angles or calculate missing angles from $180^\circ$.
- Classify by the largest angle: below, equal to, or above $90^\circ$.
- State both classifications when information permits.
Worked example
Representations and interpretation
A side–angle correspondence links the longest side to the largest opposite angle. A hierarchy shows equilateral inside isosceles under inclusive definitions, while angle categories are mutually exclusive for nondegenerate triangles.
Reasoning about variations
Side sets $2,3,5$ do not form a nondegenerate triangle because equality gives a straight segment. Classification should occur only after existence is checked.
Common mistakes
How to check your work
- Add angles to $180^\circ$.
- Apply all three triangle inequalities or the largest-side shortcut.
- Match the largest side with the largest opposite angle.
Practice
- Classify sides $7,7,7$.
- Can $4,6,11$ form a triangle?
- A triangle has angles $30^\circ,60^\circ,90^\circ$. Classify by angles.
Answers and brief solutions
Show answers
- Equilateral All three sides are congruent.
- No $4+6=10<11$.
- Right It contains one $90^\circ$ angle.
Synthesis and transfer
Use measured side lengths and the converse Pythagorean comparison to classify a survey triangle without relying on a sketch that may not be drawn to scale.
For side lengths $5$, $5$, and $8$, the triangle is isosceles by sides. Comparing the square of the longest side with the other squares gives $8^2=64>25+25=50$, so it is obtuse by angles. Both labels apply at once. Measurements must first satisfy the triangle inequality; for $2$, $3$, and $6$, no triangle exists and angle classification is meaningless. When coordinates are given, distance formula and dot products can supply exact side and angle information without trusting a sketch. Classification is strongest when its evidence is stated alongside the label.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Classify sides $7,7,7$.
- All three sides are congruent.
End of lesson
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