Question

How big is each angle in the pentagon? (1) One angle is \(100^{\circ}\) (2) Four angles are equal 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

The answer is C.

Step-by-step solution

Understand the total internal angles in a pentagon

A pentagon has 5 sides. The sum of the internal angles of any polygon is given by the formula \[ (n-2) \times 180^{\circ} \] where \( n \) is the number of sides. For a pentagon, \[ n = 5 \] so the sum of the internal angles is \[ (5-2) \times 180^{\circ} = 3 \times 180^{\circ} = 540^{\circ} \].

Use the first condition

The first condition states that one angle is \(100^{\circ}\). Since the sum of all angles in the pentagon is \(540^{\circ}\), the sum of the remaining four angles is \[ 540^{\circ} - 100^{\circ} = 440^{\circ} \].

Use the second condition

The second condition states that the other four angles are equal. Let each of these angles be \( x \). So, we have \[ 4x = 440^{\circ} \] solving for \( x \), \[ x = \frac{440^{\circ}}{4} = 110^{\circ} \].

Confirm the angles

The angles are verified as follows: one angle is \(100^{\circ}\) and four angles are \(110^{\circ}\). The sum is \[ 100^{\circ} + 4 \times 110^{\circ} = 100^{\circ} + 440^{\circ} = 540^{\circ} \], which is the total internal angle sum of a pentagon. Therefore, the conditions given are sufficient to determine the angles.

Conclusion

Both conditions (1 and 2) together are necessary to solve for the angles. Therefore, the answer is C.

Key concepts

pentagon angle sum

When dealing with regular pentagons or even irregular ones, it's crucial to know the sum of their internal angles. A pentagon is a type of polygon with 5 sides. You can determine the total sum of internal angles for any polygon using the formula \( (n-2) \times 180^{\circ} \). Here, \(n\) stands for the number of sides.
For a pentagon, \(n = 5\), so the calculation becomes:
\[ (5-2) \times 180^{\circ} = 3 \times 180^{\circ} = 540^{\circ} \].
This means that in any pentagon, if you add up all the internal angles, they will always sum up to \(540^{\circ} \). This basic fact is incredibly useful when solving various geometric problems involving pentagons.

internal angles formula

The internal angles formula is a key tool in geometry, particularly when dealing with polygons. As mentioned, the formula to calculate the sum of internal angles for any polygon is: \[ (n-2) \times 180^{\circ} \].
Here are some important points to note:

• This formula helps in identifying how angles spread within a polygon.
• It's derived from dividing the polygon into triangles, each of which has an angle sum of \(180^{\circ} \).

For example, for a hexagon (which has 6 sides), you would calculate it as:
\[ (6-2) \times 180^{\circ} = 4 \times 180^{\circ} = 720^{\circ} \].
This helps in understanding complex shapes by breaking them down into simpler parts.

solving geometric problems

Geometric problems often involve applying multiple concepts and formulas to find the solution. Here's how you can approach such problems:

• Start by understanding the given conditions and identifying what you need to find.
• Use relevant geometrical formulas and theorems. For polygons, this often involves the internal angles formula.

In the pentagon problem, you are given that one angle is \(100^{\circ}\) and the four other angles are equal. Using the total internal angle sum \(540^{\circ}\), you can set up an equation:
\[ 100^{\circ} + 4x = 540^{\circ} \]
Solving for \(x\):
\[ 4x = 540^{\circ} - 100^{\circ} = 440^{\circ} \]
\[ x = \frac{440^{\circ}}{4} = 110^{\circ} \]
Therefore, the angles are \(100^{\circ}\) and \(4 \times 110^{\circ}\). If you add them up, you get \(540^{\circ}\), confirming your answer. It's often necessary to use multiple conditions together, as seen in this problem, to arrive at the solution.