lesson
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If you rip all three corners off any paper triangle and line them up point-to-point, something surprising happens every single time.
The corners fit together perfectly to form a straight line. An interior angle is an angle formed inside a polygon where two sides meet at a vertex.
📊Interactive/animated demonstration showing a triangle with three vertices colored red (Angle A), blue (Angle B), and green (Angle C). The three corners tear away and smoothly rotate and translate downward to sit side-by-side along a flat baseline. An arc spans all three touching angles showing they form a straight line labeled 'Straight angle = 180°'. Card style with clean modern colors (#1e2945 text, #ffffff background, #e6e6e6 border, vibrant red/blue/green accents).
Because a straight line measures 180∘, the sum of the three interior angles in any triangle is always 180∘.
Around 300 BCE, the Greek mathematician Euclid proved this fact in his book Elements by drawing a parallel line through the top vertex, showing that the corner angles match alternate angles on the line.
So if two angles in a triangle are given, how do we use this 180∘ rule to find the missing one?
Finding a Missing Angle
To find an unknown angle, add the two known angles together and subtract that sum from 180∘.
📊Step-by-step visual diagram of a triangle with interior angles labeled 65°, 75°, and an unknown angle labeled 'x°'. Below the triangle, show the equation flow in crisp pill badges: Step 1: 65° + 75° = 140°. Step 2: 180° - 140° = 40°. Step 3: x = 40°. Clean layout responsive to 350px width, text color #1e2945, subtle glowing box around the final answer 40°.
We can check our work by adding all three values together: 65∘+75∘+40∘=180∘.
What happens if a triangle gives you only one angle number and a little square symbol in the corner?
Hidden Angle Clues
A square in a corner indicates a right angle, which always measures exactly 90∘ even if the number 90 is not written down.