Question

Find a counterexample to the following statement. An exterior angle measure of any convex polygon can be found by dividing 360 by the number of interior angles.

Short answer

A convex irregular quadrilateral shows not all exterior angles are 90°, countering the statement.

Step-by-step solution

Understand the Statement

The statement suggests that the measure of each exterior angle of any convex polygon can be found by dividing 360 degrees by the number of its interior angles. For clarification, this implies that if a polygon has \(n\) sides (or \(n\) interior angles), each exterior angle would be \(\frac{360}{n}\) degrees.

Recall Polygon Angle Facts

Recall that the sum of the exterior angles of any convex polygon, regardless of the number of sides, is always 360 degrees. Therefore, dividing 360 by the number of sides does give the measure of each exterior angle for regular polygons (where all angles are equal). However, this property doesn't necessarily hold for irregular polygons.

Construct a Convex Irregular Polygon

Consider a convex irregular quadrilateral, which has 4 sides. If the statement were true, each exterior angle measure would be \(\frac{360}{4} = 90\) degrees. However, the actual angle measures differ because the quadrilateral is irregular, meaning its sides and angles aren't equal.

Identify the Counterexample

In a convex irregular quadrilateral (not regular), the exterior angles are not necessarily all 90 degrees, as they vary depending on the shape's specific dimensions. Therefore, the measure of each exterior angle isn't necessarily \(\frac{360}{4}\). This irregular quadrilateral serves as a counterexample to the general statement.

Conclude the Counterexample

A specific case, such as a convex irregular quadrilateral whose exterior angles aren't all 90 degrees, contradicts the statement. Hence, the statement doesn't hold for all convex polygons, invalidating it universally.

Key concepts

Convex Polygon

A convex polygon is a type of polygon where all the interior angles are less than 180 degrees. This ensures that no part of the polygon "caves in" or is recessed.
Convex polygons have the following key characteristics:

• All the vertices (corners) point outwards.
• Any line segment drawn between two points within the polygon always lies entirely inside or on the polygon's boundaries.
• As a result of these properties, convex polygons are always simple shapes, meaning they do not intersect themselves.

The concept of convexity is vital in geometry because it simplifies many calculations and properties that govern polygons, including understanding their angles and area.

Irregular Quadrilateral

An irregular quadrilateral is a four-sided polygon that does not have all sides equal, nor all angles equal. This makes it very different from squares or rectangles, which are regular quadrilaterals.
Here’s what makes an irregular quadrilateral unique:

• It may have sides of different lengths.
• The angles can vary and are not required to sum up to equal measures.
• Despite the differences in angles and sides, the sum of the interior angles of any quadrilateral is always 360 degrees.

When constructing or analyzing irregular quadrilaterals, each shape can have completely different properties, making them interesting examples in studying polygons. Their exterior angle measures can vary widely, disproving the notion that dividing 360 by the number of sides provides each exterior angle's measure.

Exterior Angle Sum

The exterior angle sum of a polygon is a fascinating concept that gives us a standard rule: the sum of all exterior angles of any convex polygon is always 360 degrees. This property holds true no matter how many sides the polygon has or how irregular the shape might be.
Here's how it works with exterior angles:

• An exterior angle is formed by extending one side of the polygon beyond its vertex.
• By combining one exterior angle at each vertex, their total will always add up to 360 degrees, fulfilling this consistent rule.
• However, while the sum remains constant, the individual angle measures can vary greatly, particularly in irregular polygons like the irregular quadrilateral.

Understanding the sum of exterior angles is crucial in solving and validating statements about polygon properties and disproving false generalizations, as shown with irregular shapes.