Classifying triangles
By sides:
| Type | Sides | Angles |
|---|---|---|
| Equilateral | 3 equal | all 60° |
| Isosceles | 2 equal | 2 equal base angles |
| Scalene | none equal | none equal |
By angles:
| Type | Angles |
|---|---|
| Acute | all less than 90° |
| Right | one angle exactly 90° |
| Obtuse | one angle more than 90° |
A triangle can be described both ways: a "right isosceles triangle" has a 90° angle and two equal sides (its other angles are 45° each).
The angle sum
The interior angles of every triangle add to 180°. This is the angle sum property.
Try it: cut out a paper triangle, tear off the three corners and line them up. They make a straight line, 180°.
Example: Two angles are 50° and 65°. The third is 180 − 50 − 65 = 65°. (So the triangle is isosceles!)
Isosceles triangles
The two equal sides meet at the apex. The angles opposite them, the base angles, are equal.
Example: The apex angle is 40°. Base angles = (180 − 40) ÷ 2 = 70° each.
Exterior angles
Extend one side of a triangle. The angle outside, the exterior angle, equals the sum of the two interior angles opposite it.
Example: Interior angles 45° and 60° are opposite an exterior angle. The exterior angle = 45 + 60 = 105°.
(Check: the third interior angle is 75°, and 75 + 105 = 180 on the straight line ✓.)
Side and angle relationships
- The longest side is opposite the largest angle.
- The shortest side is opposite the smallest angle.
The triangle inequality
Not every three lengths make a triangle. The triangle inequality says any two sides must add to more than the third.
- 3, 4, 5: 3 + 4 = 7 > 5 ✓ (a triangle)
- 2, 3, 6: 2 + 3 = 5 < 6 ✗ (the short sides can’t reach)
Worked examples
Example 1: Find the missing angle: 90° and 34°.
180 − 124 = 56°.
Example 2: An isosceles triangle has base angles of 52°. Find the apex.
180 − 104 = 76°.
Example 3: Can 5 cm, 7 cm and 13 cm form a triangle?
5 + 7 = 12 < 13. No.
Common mistakes
- Thinking the angle sum depends on size. Every triangle’s angles add to 180°.
- Choosing the wrong angles as equal in an isosceles triangle. The equal angles are opposite the equal sides.
- Forgetting the triangle inequality.
Try it yourself
- Two angles are 72° and 48°. Find the third.
- Classify a triangle with angles 30°, 60°, 90°.
- An isosceles triangle’s apex angle is 100°. Find the base angles.
- An exterior angle is 130°. One opposite interior angle is 55°. Find the other.
- Can 4, 4 and 9 form a triangle?
Answers: 1) 60° 2) right scalene 3) 40° each 4) 75° 5) no
Quick recap
- A triangle‘s angles add to 180°.
- Classify by sides (equilateral, isosceles, scalene) and by angles.
- Exterior angle = sum of the two opposite interior angles.