Key takeaway: The exterior angle theorem says an angle formed outside a triangle by extending one side is worth the same as the two far-away inside angles combined.
Grade 88.G.A.5Geometry and Measurement
Definition of Exterior Angle Theorem
The exterior angle theorem links an outside angle of a triangle to the inside angles. Stretch one side past a vertex; the angle between that stretched side and the next side of the triangle is an exterior angle, and its measure equals the sum of the two interior angles at the other vertices (the remote interior angles).
The proof uses straight lines. The exterior angle and its neighboring interior angle fill a flat angle of \(180^{\circ}\), and the three interior angles also total \(180^{\circ}\). Take away the shared neighbor and what is left on each side must match.
Formula and notation
If the exterior angle at vertex \(C\) is called \(\angle ACD\) and the remote interior angles are at \(A\) and \(B\), the exterior angle theorem is the equation
\[m\angle ACD = m\angle A + m\angle B\]
Each triangle has six exterior angles, two at every vertex, and the two at one vertex are equal because they are vertical angles.
Key facts
- An exterior angle is always larger than either of its remote interior angles.
- Its next-door inside angle and it always make a straight angle.
- One exterior angle at each vertex gives three angles that total 360 degrees.
- The exterior angle theorem finds a missing interior angle in one step.
Exterior Angle Theorem example
In triangle \(PQR\), side \(QR\) is extended past \(R\) to point \(S\). Angle \(P\) measures \(58^{\circ}\), angle \(Q\) measures \((2y + 9)^{\circ}\), and the exterior angle \(\angle PRS\) measures \((5y – 14)^{\circ}\). Find \(y\).
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The remote interior angles for \(\angle PRS\) are \(P\) and \(Q\), so \(5y – 14 = 58 + (2y + 9)\).
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Simplify: \(5y – 14 = 2y + 67\), so \(3y = 81\) and \(y = 27\).
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Exterior angle: \(5(27) – 14 = 121\). Check: \(Q = 2(27) + 9 = 63\) and \(58 + 63 = 121\).
\(y = 27\) and the exterior angle measures \(121^{\circ}\).
Common mistake
Watch out
Ines claims the outside angle equals all three inside angles added together, so for a triangle with angles \(41^{\circ}\), \(76^{\circ}\) and \(63^{\circ}\) she gets \(180^{\circ}\) for the exterior angle at the \(63^{\circ}\) vertex.
Why it is wrong
Only the two remote interior angles count. The third interior angle is the neighbor of the exterior angle, and together they make \(180^{\circ}\). Correct value: \(180 – 63 = 117\), which equals \(41 + 76\).
Related terms and lessons
- Triangle Angle Sum Theorem
- Supplementary Angles
- Adjacent Angles
- Vertical Angles
- Alternate Interior Angles
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