DWG NO. 04.1 — Lesson 1 of 6

Classifying Triangles

Unit 4: Congruent Triangles · ~15–30 min

Objective

Classify a triangle by its side lengths and by its angle measures, and name both classifications from a figure or a set of measurements.

Two ways to sort a triangle

Every triangle can be described two independent ways: by comparing its side lengths to each other, and by looking at its largest angle. A single triangle gets one label from each column — for example, a triangle can be both right and scalene at the same time.

Classifying by sides

Type	Condition	Angle consequence
Scalene	All three sides have different lengths.	All three angles are different too.
Isosceles	At least two sides are congruent.	The angles opposite those two sides are congruent.
Equilateral	All three sides are congruent.	All three angles are congruent (each 60°).

Classifying by angles

Type	Condition
Acute	All three angles measure less than 90°.
Right	Exactly one angle measures exactly 90°.
Obtuse	Exactly one angle measures more than 90°.
Equiangular	All three angles are congruent (each 60°) — this is always also equilateral.

Notice that a triangle can never have two right angles or two obtuse angles — the three angles always sum to 180°, so at most one angle can be 90° or larger. You'll prove this fact formally in the next lesson.

Left: isosceles triangle ABC (AB ≅ BC, so ∠A ≅ ∠C). Right: right triangle DEF, marked with a right-angle box at D.

Worked Example 1 · Classify by side lengths

ProblemTriangle GHI has sides GH = 5, HI = 5, GI = 8. Classify it by its sides.

1.Compare all three lengths: 5, 5, and 8. Two of the three sides match (GH = HI = 5), and the third is different.

2.Exactly two sides congruent, none of them all three — that's the definition of isosceles, not equilateral.

Triangle GHI is isosceles.

Worked Example 2 · Classify by angle measures

ProblemTriangle JKL has angles measuring 42°, 96°, and 42°. Classify it by its angles, and note anything you can say about its sides.

1.One angle, 96°, is greater than 90°. The other two are each less than 90°. Exactly one angle over 90° makes this an obtuse triangle.

2.The two 42° angles are congruent to each other. Since congruent angles sit opposite congruent sides, the sides opposite those two angles must also be congruent — so the triangle is also isosceles.

Triangle JKL is obtuse and isosceles.

Guided practice

1.A triangle has sides 9, 9, and 9. Name both its side classification and its angle classification.
Show answer
All three sides equal → equilateral. Equilateral triangles are always equiangular, with every angle 60°.
2.Triangle MNP has angles 55°, 60°, and 65°. Classify it by angle. Can you tell if it's scalene, isosceles, or equilateral just from these angles?
Show answer
All three angles are under 90°, so it's acute. Since all three angles are different, all three sides must be different too — it's scalene.
3.Can a triangle be both right and equilateral? Explain.
Show answer
No. Equilateral means all angles equal 60°, but a right triangle needs one angle to equal 90°. A triangle can't satisfy both at once.
4.Triangle QRS has sides QR = 4, RS = 6, QS = 7. Classify it by side length. What does that tell you about its three angles?
Show answer
All three sides have different lengths, so it's scalene. That means all three angles must also be different from each other.
5.A triangle has one angle of 110°. What can you immediately rule out about its other two angles, and what type of triangle is it?
Show answer
It's obtuse, since one angle exceeds 90°. You can immediately rule out either other angle being 90° or more — a triangle can have only one angle at or above 90°, since the three angles must sum to 180°.

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