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Types of Angles and Triangles

Angles and triangles are the building blocks of geometry. Every polygon can be broken into triangles, and every shape involves angles. Getting these fundamentals right makes everything else in geometry , from area formulas to trigonometry , much clearer.

In this lesson

  1. Types of angles
  2. Angle pairs: complementary and supplementary
  3. Triangles classified by angles
  4. Triangles classified by sides
  5. Rules every triangle must follow

1 Types of Angles

Angle types by size
⟨Acute angle: Less than 90°. Sharp corner. Example: 45°, 60°, 89°.
⊾Right angle: Exactly 90°. Marked with a small square in diagrams. The corner of a piece of paper.
⟩Obtuse angle: Greater than 90° but less than 180°. Wide, blunt corner. Example: 120°, 150°.
,Straight angle: Exactly 180°. A straight line.
↺Reflex angle: Greater than 180° but less than 360°. The "large" part of an angle that goes more than halfway around.

2 Angle Pairs: Complementary and Supplementary

Complementary angles add up to 90°. If one angle is 35°, its complement is 55°. They don't need to be adjacent , they just need to sum to 90°.

Supplementary angles add up to 180°. If one angle is 110°, its supplement is 70°. Angles on a straight line are always supplementary.

Vertical angles are formed when two lines cross. The opposite angles are equal. If one angle is 40°, the angle directly across from it is also 40°, and the two adjacent angles are each 140°.

Memory trick

Complementary = Corner (90°). Supplementary = Straight line (180°). The C and S are in alphabetical order, and 90° comes before 180°.

3 Triangles Classified by Angles

Every triangle has exactly three angles that sum to 180°. Based on those angles:

Triangle types by angle
△Acute triangle: All three angles are less than 90°. Example: 60°-70°-50°.
⊾Right triangle: One angle is exactly 90°. The other two are complementary (add to 90°). The Pythagorean theorem applies.
▷Obtuse triangle: One angle is greater than 90°. A triangle can only have one obtuse angle , if one angle is 100°, the other two must sum to 80°.

4 Triangles Classified by Sides

Triangle types by side length
△Equilateral triangle: All three sides equal. All three angles are 60°. Always also acute.
△Isosceles triangle: Two sides equal. The angles opposite the equal sides are also equal.
△Scalene triangle: All three sides different lengths. All three angles are different sizes.

These classifications overlap. A right triangle can also be isosceles (a 45-45-90 triangle). An equilateral triangle is always also acute. A scalene triangle can be acute, right, or obtuse.

5 Rules Every Triangle Must Follow

Angle sum = 180°. Always. This is the most important triangle rule. If you know two angles, subtract from 180° to find the third.

Triangle inequality theorem: each side must be less than the sum of the other two sides. You can't make a triangle with sides 2, 3, and 10 , because 2 + 3 = 5, which is less than 10. The shorter sides can't reach each other to close the triangle.

Find the Missing Angle
A triangle has angles 47° and 83°. Find the third angle.
1All angles sum to 180°
2Third angle = 180° − 47° − 83° = 50°
3Check: 47 + 83 + 50 = 180 ✓
Answer: Third angle = 50° , this is an acute triangle (all angles under 90°)
A triangle can have at most one right angle or obtuse angle

If one angle is 90° or more, the other two must together be less than 90° , so they must both be acute. You can't have two right angles (90+90=180, leaving 0° for the third), and you can't have two obtuse angles.

Practice Problems

A triangle has angles 55° and 70°. What is the third angle, and what type of triangle is it?
Third angle = 180 − 55 − 70 = 55°. Two angles are equal (55°), so it's isosceles. All angles are under 90°, so it's also acute.
Two angles are complementary. One is 28°. What is the other?
Complementary angles sum to 90°. Other angle = 90 − 28 = 62°.
Can a triangle have sides 5, 5, and 11?
Check triangle inequality: 5 + 5 = 10, which is less than 11. No , this cannot form a triangle.
What type of triangle has exactly two equal sides and one angle of 90°?
An isosceles right triangle. The two equal sides are the legs, and the angles are 90°-45°-45°.

Sources & Further Reading

The explanations on this page draw on the following established sources. We link to primary and secondary sources so you can verify claims and go deeper on any topic.

Khan AcademyAngles and TrianglesVideo lessonsMath is FunTypes of TrianglesVisual guideDesmosGeometry ToolsExplore anglesWikipediaTriangleFull formal treatment