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Geometry

Triangle Angle Sum

Lesson

The three angles of any triangle always sum to 180∘180^\circ. That single fact powers most triangle problems.

Special cases

  • Equilateral: all three angles 60∘60^\circ.
  • Isosceles: the two base angles (opposite equal sides) are equal.
  • Right: one angle is 90∘90^\circ; the two acute angles sum to 90∘90^\circ.

Exterior angle theorem

An exterior angle of a triangle equals the sum of the two remote interior angles.

Triangle inequality

Each side must be less than the sum of the other two: ∣b−c∣<a<b+c|b - c| < a < b + c.

Worked example

Two angles are 60∘60^\circ and 80∘80^\circ. Third:

180−60−80=40∘180 - 60 - 80 = 40^\circ

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Triangle with 60∘ and 80∘. Third?\text{Triangle with } 60^\circ \text{ and } 80^\circ. \text{ Third?}

Problem 2

Triangle with 45∘ and 90∘. Third?\text{Triangle with } 45^\circ \text{ and } 90^\circ. \text{ Third?}

Problem 3

Each angle of an equilateral triangle\text{Each angle of an equilateral triangle}

Problem 4

Each base angle of an isosceles right triangle\text{Each base angle of an isosceles right triangle}

Practice

Standard problems matching the lesson.

Problem 5

Triangle 50∘,60∘. Third?\text{Triangle } 50^\circ, 60^\circ. \text{ Third?}

Problem 6

Triangle 30∘,100∘. Third?\text{Triangle } 30^\circ, 100^\circ. \text{ Third?}

Problem 7

Right triangle, one acute 35∘. Other acute?\text{Right triangle, one acute } 35^\circ. \text{ Other acute?}

Problem 8

Angles in ratio 1:2:3. Smallest?\text{Angles in ratio } 1:2:3. \text{ Smallest?}

Problem 9

Angles in ratio 2:3:4. Middle?\text{Angles in ratio } 2:3:4. \text{ Middle?}

Problem 10

Isosceles vertex 40∘. Each base angle?\text{Isosceles vertex } 40^\circ. \text{ Each base angle?}

Problem 11

Isosceles base 75∘. Vertex angle?\text{Isosceles base } 75^\circ. \text{ Vertex angle?}

Problem 12

Exterior angle 110∘. Sum of remote interiors?\text{Exterior angle } 110^\circ. \text{ Sum of remote interiors?}

Problem 13

Remote interiors 40∘,70∘. Exterior at third vertex?\text{Remote interiors } 40^\circ, 70^\circ. \text{ Exterior at third vertex?}

Problem 14

Triangle 25∘,100∘. Third?\text{Triangle } 25^\circ, 100^\circ. \text{ Third?}

Problem 15

Triangle with two 50∘ angles. Third?\text{Triangle with two } 50^\circ \text{ angles. Third?}

Problem 16

Hypotenuse of right triangle is opposite which angle (degrees)?\text{Hypotenuse of right triangle is opposite which angle (degrees)?}

Problem 17

Sides 6 and 7. Max integer value of third side?\text{Sides 6 and 7. Max integer value of third side?}

Problem 18

Triangle 70∘,30∘. Third?\text{Triangle } 70^\circ, 30^\circ. \text{ Third?}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

Angles 4x,5x,9x. Find x.\text{Angles } 4x, 5x, 9x. \text{ Find } x.

Problem 20

Angles (x+10),(x+20),(x+30). Find x.\text{Angles } (x+10), (x+20), (x+30). \text{ Find } x.

Problem 21

Right triangle acute 27∘. Other acute?\text{Right triangle acute } 27^\circ. \text{ Other acute?}

Problem 22

Angles x,2x,90∘. Find x.\text{Angles } x, 2x, 90^\circ. \text{ Find } x.

Problem 23

Exterior 130∘, one remote interior 50∘. Other remote?\text{Exterior } 130^\circ, \text{ one remote interior } 50^\circ. \text{ Other remote?}

Problem 24

Triangle 30∘,x,2x. Find x.\text{Triangle } 30^\circ, x, 2x. \text{ Find } x.

Problem 25

Isosceles vertex 100∘. Base angle?\text{Isosceles vertex } 100^\circ. \text{ Base angle?}

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