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.
⟨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°.