You may have heard that a triangle’s angles add to $180^\circ$. Parallel lines and a transversal show why it is true for every triangle.
Why it works
Draw a line through the top corner, parallel to the bottom side. Along that line, three angles sit side by side and make a straight angle, $180^\circ$. The left one equals the triangle’s bottom-left angle, and the right one equals the bottom-right angle (alternate interior angles). The middle one is the triangle’s top angle. So the triangle’s three angles add to $180^\circ$.
Key idea
The angles of every triangle add to $180^\circ$. An exterior angle (between one side and the extension of the next) equals the sum of the two interior angles that are not next to it.
Worked examples
Worked example: The third angle
Two angles of a triangle measure $48^\circ$ and $77^\circ$. Find the third.
- $180 - 48 - 77 = 55$.
Answer: $55^\circ$
Worked example: With algebra
A triangle’s angles are $x^\circ$, $2x^\circ$ and $3x^\circ$. Find them.
- $x + 2x + 3x = 180$, so $6x = 180$ and $x = 30$.
- The angles are $30^\circ$, $60^\circ$ and $90^\circ$.
Answer: $30^\circ$, $60^\circ$, $90^\circ$
Worked example: An exterior angle
A triangle has interior angles of $40^\circ$ and $65^\circ$. Find the exterior angle at the third corner.
- The third interior angle is $180 - 40 - 65 = 75^\circ$.
- The exterior angle makes a straight line with it: $180 - 75 = 105^\circ$.
- Shortcut: $40 + 65 = 105$.
Answer: $105^\circ$
Common mistake
Thinking bigger triangles have bigger angle sums. A huge triangle and a tiny one both have angles adding to $180^\circ$; size changes the sides, not the angle sum.