Exterior Angle of a Triangle
When one side of a triangle is extended, the angle formed outside it is an exterior angle. In NSW Year 7 maths (NESA outcome MA4‑GEO‑C‑01) you learn that it equals the total of the two interior opposite angles, the angles not next to it.
You will use this rule to find exterior and interior angles, work backwards from a known exterior angle, combine it with angles on a straight line and the angle sum, handle isosceles triangles and pronumerals, and follow a short proof of why the rule holds. It is a handy shortcut when checking angles in roof frames and bridge trusses. It is part of Year 7 Geometry.
Every question with a fully worked solution.
- Exterior Angle of a Triangle - Video - What is the Relation of an Exterior Angle of a Triangle with its Interior Angles? Watch
Theory
When one side of a triangle is extended, the angle outside the triangle is an exterior angle. It equals the sum of the two interior opposite angles (the two angles inside the triangle that are not next to it). This Year 7 maths page shows why, and how to use the rule forwards and backwards with reasons.
An exterior angle is outside the triangle, between one side and the next side extended. The interior opposite angles are the two angles inside the triangle that are not at that vertex.
The rule. If \(d^\circ\) is an exterior angle and \(a^\circ\), \(b^\circ\) are its interior opposite angles, then \(d = a + b\). Reason: exterior angle of a triangle equals the sum of the two interior opposite angles.
Why it works. \(a + b + c = 180\) (angle sum of a triangle) and \(c + d = 180\) (angles on a straight line). Both sums are \(180\), so \(a + b + c = c + d\), and taking \(c\) from both sides gives \(d = a + b\).
The angle next to it. The exterior angle and the interior angle at the same vertex lie on a straight line, so they add to \(180^\circ\).
One at each vertex. Each vertex can have its own exterior angle. With one exterior angle at each vertex, the three exterior angles add to \(360^\circ\).
The rule and the straight-line fact:
How to use the exterior angle rule
- Find the exterior angle: it is between a side and the extended next side.
- Find its two interior opposite angles: the two angles not at that vertex.
- Write exterior \(=\) sum of the two, with the reason.
- Solve: add to find the exterior angle, subtract to find an interior one.
How to check
- The exterior angle and the interior angle next to it add to \(180^\circ\).
- The three interior angles add to \(180^\circ\).
| \(x\) | \(=\) | \(72 + 45\) |
| \(\) | \(=\) | \(117\) |
\(x = 117\) (exterior angle of a triangle equals the sum of the two interior opposite angles). Check: \(\angle ACB = 180 - 117 = 63\) and \(72 + 45 + 63 = 180\).
| \(\angle DCA\) | \(=\) | \(\angle BAC + \angle CBA\) |
| \(\) | \(=\) | \(52^\circ + 63^\circ\) |
| \(\) | \(=\) | \(115^\circ\) |
\(\angle DCA = 115^\circ\) (exterior angle of a triangle).
| \(a + 58\) | \(=\) | \(130\) |
| \(a\) | \(=\) | \(72\) |
| \(b + 130\) | \(=\) | \(180\) |
| \(b\) | \(=\) | \(50\) |
\(a = 72\) (exterior angle of a triangle) and \(b = 50\) (angles on a straight line). Check: \(72 + 58 + 50 = 180\).
| \(\angle PRS\) | \(=\) | \(\angle QPR + \angle PQR\) |
| \(108^\circ\) | \(=\) | \(38^\circ + \angle PQR\) |
| \(\angle PQR\) | \(=\) | \(108^\circ - 38^\circ\) |
| \(\) | \(=\) | \(70^\circ\) |
\(\angle PQR = 70^\circ\)
| \(a + a + 44\) | \(=\) | \(180\) |
| \(2a\) | \(=\) | \(136\) |
| \(a\) | \(=\) | \(68\) |
| \(b\) | \(=\) | \(44 + 68\) |
| \(\) | \(=\) | \(112\) |
\(a = 68\) (base angles equal, angle sum \(180^\circ\)) and \(b = 112\) (exterior angle of a triangle). Check: \(68 + 112 = 180\).
| \(x + x + 24\) | \(=\) | \(136\) |
| \(2x\) | \(=\) | \(112\) |
| \(x\) | \(=\) | \(56\) |
\(x = 56\). The angles are \(56^\circ\) and \(80^\circ\). Check: \(56 + 80 = 136\).
(i) \(\angle ACD\)
(ii) \(\angle ADE\)
| \(\angle ACD\) | \(=\) | \(35^\circ + 25^\circ = 60^\circ\) |
| \(\angle ADE\) | \(=\) | \(60^\circ + 45^\circ = 105^\circ\) |
(i) \(60^\circ\) (exterior angle of \(\triangle ABC\))
(ii) \(105^\circ\) (exterior angle of \(\triangle ACD\)). Use the rule in one triangle at a time.
Common pitfalls
Frequently asked questions
What is an exterior angle of a triangle?
The angle between one side of the triangle and the next side extended, outside the triangle.
What is the exterior angle rule?
An exterior angle of a triangle equals the sum of the two interior opposite angles.
Which angles are the interior opposite angles?
The two angles inside the triangle that are not at the vertex of the exterior angle.
How do you find a missing interior angle from an exterior angle?
Subtract the known interior opposite angle from the exterior angle. If the exterior angle is \(108^\circ\) and one angle is \(38^\circ\), the other is \(70^\circ\).
What do the exterior angles of a triangle add to?
With one exterior angle at each vertex, they add to \(360^\circ\).