Three angles inside a triangle
A triangle has three corners, and at each corner the two sides that meet make an angle inside the triangle. These are the interior angles of the triangle. Call them a, b and c.
Whatever the shape of the triangle, long and thin or nearly equal on all sides, its three interior angles always add up to the same amount: 180°.
A triangle with its three interior angles marked a, b and c.
Tear off the corners
Cut a triangle out of paper and tear off its three corners. Put the three torn corners together so that their tips meet at one point, each corner next to the last. They fit side by side with no gap and no overlap, and their outer edges make a straight line.
A straight line is a half turn, 180°. So the three angles of that triangle add up to 180°: a + b + c = 180°.
The three corners of the same triangle, laid side by side at one point, fill a straight line: a + b + c = 180°.
Why it is true for every triangle
Tearing corners shows the fact for one paper triangle. Parallel lines prove it for every triangle. Draw a line through the top corner, parallel to the bottom side.
The left side of the triangle crosses the two parallel lines, so it makes a pair of alternate angles: the angle between the new line and the left side, at the top corner, is equal to a, the bottom-left angle. In the same way, the right side makes the angle between the new line and the right side equal to c, the bottom-right angle.
At the top corner there are now three angles side by side along the new straight line: a copy of a, the triangle’s own angle b, and a copy of c. Angles on a straight line add up to 180°, so a + b + c = 180°. Nothing here depends on the shape of the triangle.
the line through C parallel to AB makes the angle on its left equal to A and on its right equal to B (alternate angles), and with C between them the three fill the straight line: A + B + C = 180°
Slide C until angle C is a right angle
Here the corners are named A and B along the base and C at the top, and the dashed line through C is parallel to the base. The two angles beside C on the dashed line are alternate angles, equal to A and B. Drag C along the dashed line: the three angles change, but together they always fill the line, A + B + C = 180°.
Finding the third angle
If two angles of a triangle are known, the third is 180 minus both of them. A triangle with angles of 50° and 60° has a third angle of 180 − 50 − 60 = 70°. Check: 50 + 60 + 70 = 180.
In a right-angled triangle, one angle is 90°, so the other two add up to 180 − 90 = 90°: they are complementary. If one of them is 25°, the other is 90 − 25 = 65°.
Two slips are common. 180 − 50 = 130 takes off only one of the known angles; both come off the 180. And 50 + 60 = 110 adds the two known angles, which is how much of the 180 is used, not what is left.
The exterior angle
Make the bottom side of the triangle longer, past the bottom-right corner, where the angle c is. The angle between that extended side and the right side of the triangle lies outside the triangle. It is called an exterior angle. Call it e.
The exterior angle e and the interior angle c sit side by side on the straight bottom line, so e + c = 180, and e = 180 − c.
But the triangle’s angles also give a + b + c = 180, so a + b = 180 − c too. Both e and a + b are 180 − c, so e = a + b. An exterior angle of a triangle is equal to the two interior angles at the other two corners added together.
For the triangle with angles of 50°, 60° and 70°, the exterior angle beside the 70° corner is 180 − 70 = 110°, and the other two corners give 50 + 60 = 110 as well.
One side carried on past the corner makes the exterior angle e. e and c fill a straight line, and e = a + b.
Worked example: Exterior Angle of a Triangle
Question In triangle ABC, side BC is extended to D. ∠ ACD = 121° and ∠ ABC = 67°. Find ∠ BAC.
1.BC is extended to D, so ∠ ACD is an exterior angle of triangle ABC.
BC is extended to D, so ∠ ACD sits outside the triangle. 2.An exterior angle equals the sum of the two interior angles opposite it: ∠ ACD = ∠ BAC + ∠ ABC.
The exterior angle at C is the two interior angles away from C, added. 3.∠ BAC = 121° − 67° = 54°.
∠ BAC = 121° − 67° = 54°. 4.Check inside the triangle: ∠ ACB = 180° − 121° = 59°, from the straight line BCD.
Inside: ∠ ACB = 180° − 121° = 59°, on the straight line BCD. 5.54° + 67° + 59° = 180°.
54° + 67° + 59° = 180°.
Answer: ∠ BAC = 54°
Common mistakes
- Writing ∠ BAC = 180° − 121° − 67°: that mixes an exterior angle into the triangle's own 180°.
- Taking the exterior angle as 180° minus the opposite angle ∠ BAC; it is 180° minus the adjacent one, ∠ ACB.
Worked example: Square Inside a Right-Angled Triangle
Question Triangle ABC has a right angle at C, with ∠ CAB = 34°. A square CDEF has D on CA, F on CB and E on AB. Find ∠ AED and ∠ BEF.
1.In triangle ABC: ∠ ABC = 180° − 90° − 34° = 56°.
The right angle at C and 34° at A leave 56° at B. 2.CDEF is a square, so DE ∥ CF, that is DE ∥ CB.
DE runs parallel to CB, the side of the square along the base. 3.∠ AED and ∠ ABC are corresponding angles: ∠ AED = 56°.
F shape: ∠ AED corresponds to ∠ ABC, so it is 56°. 4.Check in triangle ADE: ∠ ADE = 90°, so ∠ AED = 90° − 34° = 56°.
Or inside triangle ADE, whose angle at D is the square’s right angle: 90° − 34° = 56°. 5.EF ∥ DC, that is EF ∥ CA, so ∠ BEF and ∠ BAC are corresponding angles: ∠ BEF = 34°.
EF runs parallel to CA, so ∠ BEF copies ∠ BAC = 34°. 6.Check along the straight line AEB: 56° + 90° + 34° = 180°.
Along AEB: 56° + 90° + 34° = 180°.
Answer: ∠ AED = 56°; ∠ BEF = 34°
Common mistakes
- Swapping the two: ∠ AED sits in the small triangle at A, so it copies the angle at B, not at A.
- Forgetting the square's right angle at E when adding along AB, and getting 56° + 34° = 90° for a straight line.