Triangles • Topic 2 of 6

Types of Triangles

Triangles are named two ways. By sides: scalene (all different), isosceles (two equal sides, and the angles opposite them are equal), and equilateral (all sides equal, every angle 60°). By angles: acute (all under 90°), right (one 90°), and obtuse (one over 90°). One rule quietly generates many GRE questions — the triangle inequality: any side must be less than the sum of the other two and greater than their difference. That constraint is exactly how the GRE asks "which of these could be the third side" or fixes an ambiguous isosceles triangle, since the picture is not to scale and cannot be measured.

Three triangles side by side: equilateral, isosceles and right-angledThree types of trianglesEquilateralIsoscelesRight

✅ Solved examples

1. An isosceles triangle has a vertex angle of 40°. Find each base angle.
The two base angles are equal and share the remaining 180 − 40 = 140°, so each = 140 / 2 = 70°.
2. One base angle of an isosceles triangle is 55°. Find the vertex angle.
Both base angles are 55°, so the vertex = 180 − (55 + 55) = 180 − 110 = 70°.
3. Two sides of a triangle are 5 and 9. Between what two values must the third side lie?
Triangle inequality: third side > 9 − 5 = 4 and < 9 + 5 = 14. So 4 < third side < 14.
4. An isosceles triangle has sides measuring 4 and 9 (each side is one of these two lengths). Find its perimeter.
The third side equals 4 or 9. A 4-4-9 triangle fails the inequality (4 + 4 = 8 < 9), so it must be 9-9-4. Perimeter = 9 + 9 + 4 = 22.

✏️ Practice — try these, take hints as needed

1. Find each interior angle of an equilateral triangle.
All three sides equal.
All three angles equal.
180 / 3.
60° each
2. An isosceles triangle has a vertex angle of 100°. Find each base angle.
Base angles are equal.
They share 180 − 100 = 80°.
Divide by 2.
40° each
3. Two sides of a triangle are 6 and 10. Give the range of possible values for the third side.
Use the triangle inequality.
Greater than 10 − 6.
Less than 10 + 6.
4 < third side < 16
4. An isosceles triangle has two equal sides of 5 and a perimeter of 18. Find the base.
Two sides are 5 each.
5 + 5 + base = 18.
base = 18 − 10.
8
5. Can a triangle have side lengths 2, 3 and 6?
Check the triangle inequality.
Is 2 + 3 greater than 6?
5 is not greater than 6.
No — 2 + 3 < 6

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Angle Sum & Exterior Angle The Pythagorean Theorem →