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Exterior Angle Theorem: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Exterior Angle TheoremExterior Angle Theorem: Definition and ExamplesTable of ContentsExterior Angle Theorem Definition of Exterior Angle Theorem

The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the two remote interior angles. When we extend any side of a triangle, an exterior angle is formed between the extended side and the adjacent side of the triangle. The remote interior angles (also called opposite interior angles) are the angles that are non-adjacent to the exterior angle.

The Exterior Angle Inequality Theorem is a related concept that states the measure of any exterior angle of a triangle is greater than each of the opposite interior angles. This inequality theorem applies to all six exterior angles that can be formed in a triangle (two exterior angles at each vertex).

Examples of Exterior Angle Theorem Example 1: Finding a Missing Angle Using the Exterior Angle Theorem Problem:

In triangle ABC, side AB is extended to point D, forming an exterior angle ∠CAD∠CAD∠CAD.
If ∠CAD=120°∠CAD = 120°∠CAD=120° and ∠ABC=40°∠ABC = 40°∠ABC=40°, what is the measure of ∠ACB∠ACB∠ACB?

Exterior Angle Theorem

Step-by-step solution:

Step 1, Identify the exterior angle. ∠CAD∠CAD∠CAD is the exterior angle at vertex A of triangle ABC.

Step 2, Recall the Exterior Angle Theorem:

The exterior angle is equal to the sum of the two remote interior angles.

Step 3, Set up the equation:

∠CAD=∠ABC+∠ACB\angle CAD = \angle ABC + \angle ACB∠CAD=∠ABC+∠ACB

Step 4, Substitute known values:

120∘=40∘+∠ACB120^\circ = 40^\circ + \angle ACB120∘=40∘+∠ACB

Step 5, Solve for ∠ACB:

∠ACB=120∘−40∘=80∘\angle ACB = 120^\circ - 40^\circ = 80^\circ∠ACB=120∘−40∘=80∘

Step 6, Therefore, ∠ACB=80°∠ACB = 80°∠ACB=80°.

Example 2: Solving for a Variable in an Angle Equation Problem:

Find the value of xxx in the triangle where one exterior angle measures (8x+25)°(8x + 25)°(8x+25)° and its remote interior angles measure (2x+10)°(2x + 10)°(2x+10)° and (5x+20)°(5x + 20)°(5x+20)°.

Exterior Angle Theorem

Step-by-step solution:

Step 1, Identify that (8x+25)°(8x + 25)°(8x+25)° is the exterior angle of the triangle with remote interior angles (2x+10)°(2x + 10)°(2x+10)° and (5x+20)°(5x + 20)°(5x+20)°.

Step 2, Apply the Exterior Angle Theorem to set up an equation. The exterior angle equals the sum of its remote interior angles.

(8x+25)°=(2x+10)°+(5x+20)°(8x + 25)° = (2x + 10)° + (5x + 20)°(8x+25)°=(2x+10)°+(5x+20)°

Step 3, Simplify the right side of the equation.

(8x+25)°=(2x+5x+10+20)°=(7x+30)°(8x + 25)° = (2x + 5x + 10 + 20)° = (7x + 30)°(8x+25)°=(2x+5x+10+20)°=(7x+30)°

Step 4, Solve the resulting equation for xxx.

8x+25=7x+308x + 25 = 7x + 308x+25=7x+30 8x−7x=30−258x - 7x = 30 - 258x−7x=30−25 x=5x = 5x=5 Example 3: Using Exterior Angles in a Complex Triangle Problem Problem:

Find the value of ∠PRQ\angle PRQ∠PRQ using the exterior angle theorem when two exterior angles of the triangle measure 110°110°110° and 135°135°135°.

Exterior Angle Theorem

Step-by-step solution:

Step 1, Understand what we know. We have a triangle PQR with two exterior angles: ∠TQP=110°\angle TQP = 110°∠TQP=110° and ∠SPR=135°\angle SPR = 135°∠SPR=135°.

Step 2, Since QS is a straight line, we can find ∠QPR\angle QPR∠QPR using linear pair angles.

∠SPR+∠QPR=180°\angle SPR + \angle QPR = 180°∠SPR+∠QPR=180° 135°+∠QPR=180°135° + \angle QPR = 180°135°+∠QPR=180° ∠QPR=45°\angle QPR = 45°∠QPR=45°

Step 3, Apply the Exterior Angle Theorem to the exterior angle ∠TQP\angle TQP∠TQP. We know that:

∠TQP=∠PRQ+∠QPR\angle TQP = \angle PRQ + \angle QPR∠TQP=∠PRQ+∠QPR

Step 4, Substitute the known values and solve for ∠PRQ\angle PRQ∠PRQ.

110°=∠PRQ+45°110° = \angle PRQ + 45°110°=∠PRQ+45° ∠PRQ=110°−45°=65°\angle PRQ = 110° - 45° = 65°∠PRQ=110°−45°=65° Comments(3)MMechanicTomNovember 5, 2025I've used this exterior angle theorem def for my kid's studies. It's super clear & the examples really helped them grasp the concept!

DDrummerPennyNovember 4, 2025This glossary page on the exterior angle theorem is a great resource! It helped my students grasp the concept easily with its clear examples. Thanks!

NNatureLover2025September 17, 2025I’ve been using the Exterior Angle Theorem page to help my kids with their triangle homework, and it’s been a game-changer! The examples are super clear and make the concept easy to understand.

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