The three interior angles of any triangle always sum to exactly 180°, so knowing two angles determines the third.
Tear off the three corners of any paper triangle and line them up—they always form a straight line (180°). No matter how pointy or flat the triangle is, the angles always add up the same way, like three puzzle pieces that always complete a half-turn.
Showing a random 20 of 50 problems.
Example 1
easy
Can a triangle have angles 80°, 80°, and 20°? If so, classify it.
Example 2
easy
Can a triangle have angles 90°, 45°, and 45°? If so, what is it called?
Example 3
easy
Can a triangle have two right angles?
Example 4
medium
Give a one-line argument for why the angle sum is 180°, using a line through one vertex parallel to the opposite side.
Example 5
medium
What is the sum of the interior angles of a quadrilateral, derived using triangles?
Example 6
easy
A right triangle has an acute angle of 22°. Find the other acute angle.
Example 7
medium
Two parallel lines are cut by a transversal forming a triangle with a third line. One angle is 65° (alternate interior) and another is 50°. Find the third angle of the triangle.
Example 8
easy
An isosceles triangle has two base angles of 55° each. Find the vertex angle.
Example 9
easy
What is the measure of each angle in an equilateral triangle?
Example 10
hard
In △ABC, ∠A=2x+10°, ∠B=3x−5°, ∠C=x+15°. Find all three angles.
Example 11
easy
In a triangle, two angles add to 135°. What is the third angle?
Example 12
easy
What is the sum of the interior angles of any triangle?
Example 13
medium
The angles of a triangle are x, x+20°, and x+40°. Find each angle.
Example 14
easy
Can a triangle have angles of 90°, 91°, and 1°? Justify your answer.
Example 15
hard
In △ABC, the angle bisector from A meets BC at D. If ∠B=70° and ∠C=50°, find ∠ADB.
Example 16
medium
A triangle has angles x, 2x, and 3x. What kind of triangle is it?
Example 17
medium
Why can a triangle have at most one obtuse angle?
Example 18
challenge
A triangle has angles A, B, C with A:B=2:3 and C=80°. Find A and B.C = 80°, A:B = 2:3; find angles A = x and B = y.
Example 19
medium
In an isosceles triangle, each base angle is 5° more than twice the vertex angle. Find all three angles.
Example 20
challenge
On the surface of a sphere, a triangle can have three right angles (sum 270°). Why doesn't this violate the angle-sum theorem?