Problem 1 What is the sum of the measures ... [FREE SOLUTION] (2024)

Expert verified

Exterior angles are the angles formed between any side of a polygon and the extension of its adjacent side. For any polygon, whether it is a triangle, square, pentagon, or decagon, these angles are always measured outside the polygon.
When discussing polygons, it is essential to know that these figures are named based on the number of their sides. For example:

• A triangle has 3 sides.
• A quadrilateral has 4 sides.
• A pentagon has 5 sides.
• ...and so on, up to the decagon with 10 sides.

No matter how many sides a polygon has, the sum of its exterior angles always remains consistent, thanks to a specific theorem in geometry. This consistency helps simplify many geometric calculations.

Geometry theorems are fundamental rules in mathematics that establish relationships between various geometric figures and their properties. One popular and widely used theorem is the Exterior Angles Sum Theorem.
This theorem specifically asserts that for any polygon, the sum of its exterior angles is always 360 degrees. This rule applies universally to all polygons—whether regular or irregular.

• Regular polygons have all sides and interior angles equal. Examples include an equilateral triangle and a square.
• Irregular polygons have sides and angles that are not necessarily equal, like a scalene triangle or a random quadrilateral.

Despite this difference, applying the Exterior Angles Sum Theorem guarantees that the sum of the exterior angles will always equal 360 degrees. It is essential to grasp these foundational theorems, as they are key to solving more complex geometric problems.

The sum of exterior angles of a polygon is a simple yet powerful concept. According to the Exterior Angles Sum Theorem, the total sum of the measures of the exterior angles for any polygon is always 360 degrees, no matter the number of sides.
Let's delve into the application:

• For a triangle (3 sides), the sum is 360 degrees.
• For a square (4 sides), it's still 360 degrees.
• Even for a decagon (10 sides), as shown in the exercise, the sum remains 360 degrees.

This holds true because the exterior angles of any polygon form a complete circle when put together, and a circle always measures 360 degrees.
This theorem simplifies many geometric calculations and provides a reliable method to check the accuracy of other geometrical properties.
Remember, whether you're working with a simple triangle or a complex decagon, the sum of the exterior angles is a constant 360 degrees, aiding in accurate and efficient problem solving.