Geometry
GradesGrade 8GeometryExterior Angle Theorem

Exterior Angle Theorem

📐 Geometry
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Exterior Angle Theorem

An exterior angle of a triangle equals the sum of its two non-adjacent interior angles!

An EXTERIOR ANGLE of a triangle is formed by extending one side. The Exterior Angle Theorem: any exterior angle of a triangle equals the sum of the two NON-ADJACENT interior angles (called "remote interior angles"). This follows directly from the triangle angle sum of 180°.

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Exterior angle = sum of the two remote (non-adjacent) interior angles.

∠ext=∠1+∠2(remote interior angles)\angle_\text{ext} = \angle_1 + \angle_2 \quad \text{(remote interior angles)}
🔺Finding an exterior angle
Interior angles of a triangle: 55° and 72° and 53°.
Extend the side at the 53° vertex.

Exterior angle = 55° + 72° = 127°.
Check: 127° + 53° = 180° (they form a straight line) ✓

Solving exterior angle problems:

1Identify the exterior angle at the chosen vertex
2Identify the TWO remote interior angles (the ones NOT at that vertex)
3Set up: exterior angle = remote angle 1 + remote angle 2
4Solve for the unknown

The relationship at a glance:

∠a and ∠b = remote interior angles∠ext = exterior angle at vertex C∠ext = ∠a + ∠b∠ext > ∠a and ∠ext > ∠b (always!)

The exterior angle is always LARGER than either remote interior angle.

✏️ Try It!

Two remote interior angles of a triangle are 42° and 63°. What is the exterior angle at the third vertex?

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Remember This!

The exterior angle is always GREATER than either of the two remote interior angles individually. Use this as a quick check — if your answer is smaller than a remote angle, you made an error.

Practice with CalcVerse
📐Angle Relationships
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