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Exterior Angle Theorem
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Geometry Hub / Triangles / Exterior Angle Theorem
04.18 • Triangles

Exterior Angle Theorem

Study the exterior angle theorem through one expanded side of a triangle and see how an outside angle links directly to the two remote interior angles.

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△ Exterior Angle Theorem
Interactive diagram

Exterior Angle Theorem Diagram

Watch the exterior angle and the two non-adjacent interior angles so their sum relationship stays visible.

Use the movable diagram to see what defines exterior angle theorem, how the labels relate to the figure, and what stays true as the board changes.

Definition: A triangle's exterior angle equals the sum of the two remote interior angles.
Detailed definition

Understanding Exterior Angle Theorem

Exterior Angle Theorem states that an exterior angle of a triangle equals the sum of the two remote interior angles. A triangle's exterior angle equals the sum of the two remote interior angles. The remote interior angles are the two angles not adjacent to the chosen exterior angle.

This theorem is built from simple angle structure. The exterior angle and its adjacent interior angle form a linear pair, while the three interior angles of the triangle sum to one hundred eighty degrees. Together those facts produce the theorem.

It is especially useful because it turns one outside angle into information about two inside angles at once. That makes it a common step in proofs, missing-angle problems, and triangle reasoning.

Key facts

Important ideas to remember

  • A triangle's exterior angle equals the sum of the two remote interior angles.
  • The remote interior angles are the two triangle angles not touching the exterior angle.
  • An exterior angle is greater than either one remote interior angle by itself because it equals their sum.
  • The theorem depends on choosing the correct exterior angle at a side extension, not just any angle near the triangle.
Where it is used

Where exterior angle theorem shows up

  • Use the exterior angle theorem in missing-angle problems and geometric proofs.
  • Use it when a triangle side has been extended and the outside angle is labeled.
  • Use it to compare one exterior angle with two remote interior measures quickly.
Common mistakes

What to watch out for

  • Do not add the exterior angle to the adjacent interior angle as though that were the theorem; that pair forms a linear pair instead.
  • Do not use the adjacent interior angle as one of the remote angles.
  • Do not forget that the exterior angle comes from extending a side of the triangle.
Worked examples

Exterior Angle Theorem examples

Use these worked examples to see the idea in a clean diagram first, then in the kind of reasoning students usually need for classwork, homework, or test practice.

Example 1

Example 1: Reading exterior angle theorem from the diagram

Match the wording of the rule to the exact triangle parts shown before you start calculating.

  • Identify the relevant side or angle labels.
  • Compare the figure with the theorem statement.
  • Confirm that the condition for the rule is satisfied.

Result: The rule is applied to the right part of the diagram from the start.

Example 2

Example 2: Using exterior angle theorem in a solved step

Treat the theorem as one step in a complete argument and keep checking that the answer still matches the triangle.

  • Choose the needed side or angle.
  • Apply the rule carefully.
  • Test the result against the geometry of the figure.

Result: The solution remains grounded in the triangle instead of becoming a detached formula exercise.

For

Why this page helps

This page helps because the exterior angle theorem is often easier to remember visually than verbally. Once students see which two interior angles are remote, the equation becomes much less likely to be misapplied.

Do

What you can do here

  • See the exterior angle placed beside the two remote interior angles it matches in sum.
  • Compare the theorem visually with the side extension that creates the outside angle.
  • Save a clear exterior-angle-theorem diagram for notes or teaching.
Learning outcome

What this page helps you do

These takeaways are meant to help you recognize the idea faster, read diagrams more accurately, and use the topic with more confidence in real problems.

1

Exterior Angle Theorem

Apply the exterior angle theorem with better angle selection.

2

Exterior Angle Theorem

Separate remote interior angles from the adjacent interior angle more accurately.

3

Exterior Angle Theorem

Use triangle angle relationships more confidently in proofs and calculations.

04

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