Triangle angle sum and exterior angle theorems
Triangle Angle Sum and Exterior Angle
Triangle Angle Sum and Exterior Angle is the opening topic of Triangle Theorems in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.6(D), which requires students to verify theorems about the sum of the interior angles of a triangle and the exterior angle theorem.
The interior angles of a triangle sum to \(180^\circ\), and an exterior angle equals the sum of the two remote interior angles.
Theory
Two theorems govern the angles of every triangle:
- Triangle Angle Sum: the three interior angles add to \(180^\circ\).
- Exterior Angle Theorem: an exterior angle equals the sum of the two remote (non-adjacent) interior angles.
An exterior angle is formed by extending one side of the triangle; it is supplementary to the interior angle next to it.
The two theorems:
How to find a triangle angle
- Angle sum: subtract the two known angles from \(180^\circ\).
- Exterior angle: add the two remote interiors, or subtract to find a missing remote interior.
- Algebra: set the sum of angle expressions equal to \(180^\circ\) and solve.
The three angles sum to \(180^\circ\).
| \(180^\circ-70^\circ-50^\circ\) | \(=\) | \(60^\circ\) |
The exterior angle equals the sum of the two remote interior angles.
| \(40^\circ+75^\circ\) | \(=\) | \(115^\circ\) |
Set their sum equal to \(180^\circ\).
| \(x+2x+3x\) | \(=\) | \(180\) |
| \(6x\) | \(=\) | \(180\) |
| \(x\) | \(=\) | \(30\) |
The angles are \(30^\circ,60^\circ,90^\circ\).
Exterior \(=\) sum of remote interiors, so subtract.
| \(120^\circ-45^\circ\) | \(=\) | \(75^\circ\) |
Common pitfalls
Frequently asked questions
What do the angles of a triangle add up to?
Always \(180^\circ\), for every triangle.
What is the exterior angle theorem?
An exterior angle of a triangle equals the sum of the two remote (non-adjacent) interior angles.
What is a remote interior angle?
An interior angle not adjacent to a given exterior angle — the two that the exterior angle equals the sum of.
How do you find a missing angle in a triangle?
Subtract the two known interior angles from \(180^\circ\), or use the exterior angle theorem.