Question

How many sides does a regular polygon have? (1) The sum of its exterior angles \(=360^{\circ}\) (2) The total sum of all the interior and exterior angles \(=1,800^{\circ}\) A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

B. 2 alone, not 1 alone

Step-by-step solution

Analyze Exterior Angles

The sum of the exterior angles of any polygon is always 360°. For a regular polygon, all exterior angles are equal. Let the number of sides be denoted as n. Therefore, each exterior angle is given by: \[ \text{Exterior Angle} = \frac{360^\text{°}}{n} \]

Use Interior and Exterior Angles Information

For any polygon, the sum of each interior angle and its corresponding exterior angle is 180°. Given in the problem, the total sum of all interior and exterior angles combined is 1800°. Since each of these pairs sums up to 180°, we have: \[ n \times 180^\text{°} = 1800^\text{°} \]

Solve for n Using Total Sum of Angles

From the equation derived in Step 2: \[ n \times 180^\text{°} = 1800^\text{°} \] Divide both sides by 180°: \[ n = \frac{1800^\text{°}}{180^\text{°}} = 10 \]

Analyze the Sufficiency of Statements

Statement 1 tells us the sum of the exterior angles is 360°, which applies to all polygons and doesn't determine n uniquely.Statement 2 alone allowed us to find n directly. Therefore, Statement 2 alone is sufficient.

Key concepts

Polygon

To start, let's discuss what a polygon is. In geometry, a polygon is a shape with straight sides. Polygons can have different numbers of sides and angles.
Common examples include triangles, quadrilaterals, and pentagons.
Each side of a polygon connects to two other sides at a vertex, and the sides form the closed boundary of the shape.
The number of sides in a polygon is denoted by the variable \(n\). Understanding this basic definition helps in grasping more advanced concepts such as angles and their sums in polygons.

Exterior Angles

Exterior angles in a polygon are the angles formed when one side of a polygon is extended. Each exterior angle is adjacent to an interior angle.
An important fact is that the sum of all exterior angles of any polygon is always 360°.
This is true regardless of the number of sides the polygon has. In a regular polygon, where all sides and angles are equal, each exterior angle can be calculated using the formula: \[\text{Exterior Angle} = \frac{360°}{n}\]
For example, if a polygon has 6 sides (hexagon), each exterior angle would be 60°.

Interior Angles

Interior angles are the angles on the inside of a polygon. Each pair of an interior and its corresponding exterior angle add up to 180°.
For any polygon, the measure of interior angles depends on the number of sides \(n\).
The formula to find the sum of all interior angles in a polygon is given by: \[ \text{Sum of Interior Angles} = (n - 2) \times 180° \]
This means, for a polygon with 6 sides (hexagon), the sum of all interior angles would be 720°.

Sum of Angles

For the exercise in question, it was crucial to recognize the relationship between interior and exterior angles.
The sum of the exterior angles is always 360°. Moreover, the total sum of all interior and exterior angles combined for any polygon can provide useful information.
In a regular polygon where all angles and sides are equal, knowing either the sum of exterior angles or the combined sum of all angles can help determine the number of sides \(n\).
By using the provided conditions: \[ n \times 180° = 1800° \] We can find \(n\) by dividing 1800° by 180°, resulting in a polygon with 10 sides.