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Exterior angle theorem
 
 
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The exterior angle theorem is a theorem
Theorem

In mathematics, a theorem is a statement Mathematical proof on the basis of previously accepted or established statements such as axioms.In formal mathematical logic, the concept of a theorem may be taken to mean a formula that can be formal proof according to the deductive system of a fixed formal system....
 in elementary geometry which states that the exterior angle of a triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
 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.






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The exterior angle theorem is a theorem
Theorem

In mathematics, a theorem is a statement Mathematical proof on the basis of previously accepted or established statements such as axioms.In formal mathematical logic, the concept of a theorem may be taken to mean a formula that can be formal proof according to the deductive system of a fixed formal system....
 in elementary geometry which states that the exterior angle of a triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
 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.



Proof

Given: In ?ABC, angle ACD is the exterior angle.

To prove: mACD = mABC + mBAC (here, mACD denotes the size of the angle ACD)

Proof:
Statements Reason
In ?ABC, ma + mb + mc = 180°------[1] Sum of the measures of all the angles of a triangle is 180°
Also, mb + md = 180°-------[2] Linear pair axiom
? ma + mc + mb = mb + md From [1] and [2]
? ma + mc + mb = mb + md 
? md = ma + mc
i.e. mACD = mABC + mBAC 


Hence, proved.