Understanding the Exterior Angles Theorem | Exploring the Relationship Between Exterior and Interior Angles of Triangles

Exterior Angles Theorem

The Exterior Angles Theorem relates the measure of an exterior angle of a triangle to the measures of its two remote interior angles

The Exterior Angles Theorem relates the measure of an exterior angle of a triangle to the measures of its two remote interior angles.

In a triangle, an exterior angle is formed when one side of the triangle is extended. The exterior angle is the angle formed between this extended side and the adjacent side of the triangle.

According to the Exterior Angles Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. “Remote interior angles” refers to the two non-adjacent interior angles of a triangle with respect to the exterior angle.

Mathematically, if we denote the measures of the three angles of a triangle as A, B, and C, and the measure of the exterior angle as E, then the Exterior Angles Theorem can be written as:
E = A + B, or
E = B + C, or
E = A + C

This theorem is particularly useful when solving problems involving angles in triangles, as it allows us to find the measure of an unknown angle by using the measures of the known angles.

To illustrate with an example: Let’s say we have a triangle with interior angles A, B, and C measuring 45 degrees, 60 degrees, and x degrees respectively. If we extend side AB, creating an exterior angle, we can use the Exterior Angles Theorem to find the measure of this exterior angle. Since the exterior angle is adjacent to angles A and B, we can write the equation:
x = 45 + 60
Simplifying, we find that x = 105.

Thus, the Exterior Angles Theorem helps us understand and calculate the relationships between the exterior and interior angles of a triangle, providing a valuable tool when solving problems involving triangles and their angles.

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