Question

In the adjoining figure, a star is shown. What is the sum of the angles \(A, B, C, D\) and \(E ?\) (a) \(120^{\circ}\) (b) \(180^{\circ}\) (c) \(240^{\circ}\) (d) can't be determined

Short answer

Answer: (d) can't be determined

Step-by-step solution

Determine the number of sides of the polygon

To find the number of sides of the star-shaped polygon, let's count from one angle (e.g., A) to another (e.g., B), moving clockwise along each vertex. We find that there are five distinct vertices, and thus the polygon has five sides.

Apply the sum of angles formula

Now that we know the polygon has 5 sides, we can apply the formula to find the sum of all interior angles: \((n-2) \times 180^{\circ}\). \(n = 5\) (number of sides) Plugging this into the formula: \((5 - 2) \times 180^{\circ} = 3 \times 180^{\circ} = 540^{\circ}\) So, the sum of all the angles of this polygon is \(540^{\circ}\).

Find the sum of angles A, B, C, D, and E

The sum of the angles of the star-shaped polygon is formed by the angles A, B, C, D, and E along with 5 other supplementary angles. Each of these supplementary angles is \(180^{\circ} - X\), where X represents the corresponding angle A, B, C, D, or E. Therefore, the sum of all the angles of the polygon is equal to the sum of angles A, B, C, D, and E plus the sum of these 5 supplementary angles. Let S be the sum of angles A, B, C, D, and E. \(540^{\circ} = S + 5(180^{\circ} - X)\), where X is the corresponding angle A, B, C, D, or E. Now, we see that sum of all supplementary angles is \(5(180^{\circ} - X) = 900^{\circ} - 5X\). Therefore, \(540^{\circ} = S + (900^{\circ} - 5X)\) Rearranging this equation, we have: \(S = 540^{\circ} - (900^{\circ} - 5X)\) We know that \(5X\) is equal to the sum of A, B, C, D, and E (since X is a corresponding angle), so we can substitute S for 5X: \(5X = 540^{\circ} - (900^{\circ} - 5X)\) Expanding and simplifying the equation: \(5X = 540^{\circ} - 900^{\circ} + 5X\) \(5X - 5X = -360^{\circ}\) The sum of angles A, B, C, D, and E is not a valid answer because it results in a negative value, which is not possible for a sum of angles. Therefore, the correct answer is: (d) can't be determined

Key concepts

Sum of Polygon Angles

Calculating the sum of a polygon's angles is a fundamental topic in geometry. To find this sum for any polygon, we need to use the formula: \[(n-2) \times 180^{\circ}\]where \(n\) is the number of sides of the polygon. This formula works for all simple polygons, meaning ones that don't intersect themselves, like triangles, quadrilaterals, etc. For instance, a triangle \((n=3)\) would have a total interior angle sum of \((3-2) \times 180^{\circ} = 180^{\circ}\). For a quadrilateral \((n=4)\), the sum would be \((4-2) \times 180^{\circ} = 360^{\circ}\). The formula tells us how each additional side contributes \(180^{\circ}\) to the total sum of interior angles. This helps us predict the geometric properties of any given polygon quickly and accurately.

Interior Angles

Interior angles in a polygon are the angles formed between any two adjacent sides. For a simple convex polygon, all these interior angles point inwards, towards the polygon's center, and their sum can be calculated using the previously mentioned formula, \((n-2) \times 180^{\circ}\). Here's what's essential to remember about polygon interior angles:

• Each angle must be less than \(180^{\circ}\) in a convex polygon.
• The simplicity of a polygon can ensure that the interior angles, when summed, form predictable patterns based on the number of sides.

This concept is pivotal in understanding geometric transformations, such as rotation and reflection, as they rely on these angle properties.

Star-shaped Polygon

A star-shaped polygon is a fascinating shape to analyze in geometry due to its unique properties. Unlike simple polygons, star polygons have both interior and exterior segments that intersect. Star polygons can be described by their vertex configuration, known as skipping steps, that determine the intersection pattern. Even though a star-shaped polygon is more complex, it still forms a closed, symmetric shape. The special thing about these polygons is that, despite their intricate design, the sum of angles around a point, like the center, follows the same angle rules for polygons, with interior and exterior angles adding layers of calculation. This is why determining specific angle sums like those in the exercise from a star-shaped polygon can sometimes lead to issues, such as indeterminacy.

Supplementary Angles

Supplementary angles refer to two angles whose measures add up to \(180^{\circ}\). These angles are a constant presence in geometric diagrams and calculations. For example, if a polygon has an angle version that extends linearly, such as angles in a star shape, the exterior angle can be considered supplementary to the corresponding interior angle.Understanding how supplementary angles work requires noting:

• If one angle is known, everything supplementary about it can be deduced.
• Supplementary angles often appear in polygons where extended lines form reasoning for segment calculations.

In star polygons, the concept of supplementary angles becomes crucial when determining the complete set of angles, given that the non-interior angles contribute to the overall shape and geometry calculations.