Exterior angle theorem
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.
Proof
Given: In ∆ABC, angle ACD is the exterior angle.
To prove: m'ACD = m'ABC + mBAC (here, mACD denotes the size of the angle ACD)
Proof:
| Statements | Reason |
|---|---|
| In ∆ABC, m'a + m'b + m'c = 180°------[1]''' | Sum of the measures of all the angles of a triangle is 180° |
| Also, m'b + m'd = 180°-------[2] | Linear pair axiom |
| ∴ m'a + m'c + m'b = m'b + md | From [1] and [2] |
| ∴ m'a + m'c + | |
| ∴ m'd = m'a + mc | |
| i.e. m'ACD = m'ABC + mBAC |
Hence, proved.
References
- .
This article is licensed under the GNU Free Documentation License.
Last updated on Tuesday September 23, 2008 at 04:39:39 PDT (GMT -0700)
View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation
|
|
