The **exterior angle theorem** is a theorem in elementary geometry which states that the exterior angle of a triangle is equal to the sum of the two remote interior angles.

A triangle has three corners, called *vertices*. The sides of a triangle that come together at a vertex form an angle. This angle is called the *interior angle*. In the picture below, the angles *a*, *b* and *c* are the three interior angles of the triangle. An *exterior angle* is formed by extending one of the sides of the triangle; the angle between the extended side and the other side is the exterior angle. In the picture, angle *d* is an exterior angle.

The exterior angle theorem says that the size of an exterior angle at a vertex of a triangle equals the sum of the sizes of the interior angles at the other two vertices of the triangle. So, in the picture, the size of angle *d* equals the size of angle *a* plus the size of angle *c*.

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**Given:** In ∆*ABC*, angle *ACD* is the exterior angle.

**To prove:** *ACD* = *ABC* + BAC (here, *ACD* denotes the size of the angle *ACD*)

**Proof:**

Statements | Reason |
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In ∆*ABC*, *a* + *b* + *c* = 180°**------[1]** | Sum of the measures of all the angles of a triangle is 180° |

Also, *b* + *d* = 180°**-------[2]** | Linear pair axiom |

∴ *a* + *c* + *b* = *b* + *d* | From **[1]** and **[2]** |

∴ *a* + *c* + *b* = *b* +*m**d* | |

∴ *d* = *a* + *c* |

i.e. *ACD* = *ABC* + *BAC* | |

**Hence, proved.**

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