Are the statements Always, Sometimes or Never. Please explain why. Thanks.
1. An equilateral polygon is regular.
2. The greater the number of sides of a regular polygon, the smaller the measure of any of its exterior angle.
3. The sum of the measures of the interior angles of a polygon is a multiple of 360.
1. "equilateral" means equal sides, but says nothing about its angles. How is regular defined ?
2. give yourself an example,
e.g. consider a hexagon and an octogon, and see what happens
3. but the formula says 180(n-2) as the sum, so ....
e.g. take a heptagon, n = 7, so is 180(7) divisible by 360 ?
1. Always: An equilateral polygon is a special type of regular polygon. In an equilateral polygon, all sides and angles are congruent, which is one of the defining characteristics of a regular polygon. Therefore, every equilateral polygon is regular.
2. Sometimes: The statement is true for regular polygons, but not for all polygons. In a regular polygon, all exterior angles are congruent, so as the number of sides increases, the measure of each exterior angle decreases. However, for irregular polygons, it is not necessarily true. Irregular polygons can have exterior angles of varying measures.
3. Always: The sum of the measures of the interior angles of any polygon is always a multiple of 180 degrees. This is known as the Polygon Interior Angle Sum Theorem. Each interior angle is formed by extending one side of the polygon and the adjacent sides, creating a linear pair of angles, which together sum up to 180 degrees. Since a polygon has a finite number of sides, the sum of all its interior angles will be a multiple of 180 degrees. Since 180 is a divisor of 360, the sum of the interior angles will also be a multiple of 360.
1. Always. An equilateral polygon is a polygon with all sides of equal length. A regular polygon is a polygon with both equal sides and equal angles. Since an equilateral polygon satisfies both conditions, it is always regular.
2. Sometimes. The measure of an exterior angle of a regular polygon is always the same, regardless of the number of sides. However, the statement suggests that as the number of sides increases, the measure of each individual exterior angle decreases. This is not necessarily true. In fact, it is the opposite. As the number of sides of a regular polygon increases, the measure of each exterior angle increases. For example, a regular triangle (equilateral) has exterior angles of 120 degrees, while a regular hexagon has exterior angles of 60 degrees. So, the statement is sometimes true for regular polygons with more than three sides, but not always.
3. Always. The sum of the measures of the interior angles of a polygon can be found using the formula (n-2) * 180, where n represents the number of sides of the polygon. For example, a triangle has 3 sides, so the sum of its interior angles is (3-2) * 180 = 180 degrees. A quadrilateral has 4 sides, so the sum of its interior angles is (4-2) * 180 = 360 degrees. In general, the sum of the interior angles of any polygon is always a multiple of 360. Therefore, the statement is always true.