Four interior angles of a pentagon measure $88^{\circ}, 118^{\circ}, 132^{\circ}$, and $100^{\circ}$. What is the measure of the fifth interior angle?
Answer (1)
The sum of the interior angles of a pentagon is 54 0 ∘ .
Add the four given angles: 8 8 ∘ + 11 8 ∘ + 13 2 ∘ + 10 0 ∘ = 43 8 ∘ .
Subtract the sum of the given angles from 54 0 ∘ to find the fifth angle: 54 0 ∘ − 43 8 ∘ = 10 2 ∘ .
The measure of the fifth interior angle is 10 2 ∘ .
Explanation
Problem Analysis We are given four interior angles of a pentagon: 8 8 ∘ , 11 8 ∘ , 13 2 ∘ , and 10 0 ∘ . We need to find the measure of the fifth interior angle.
Sum of Interior Angles The sum of the interior angles of a polygon with n sides is given by the formula ( n − 2 ) × 18 0 ∘ . For a pentagon, n = 5 , so the sum of the interior angles is ( 5 − 2 ) × 18 0 ∘ = 3 × 18 0 ∘ = 54 0 ∘ .
Setting up the Equation Let x be the measure of the fifth interior angle. We can set up the equation: 8 8 ∘ + 11 8 ∘ + 13 2 ∘ + 10 0 ∘ + x = 54 0 ∘ .
Sum of Known Angles First, we calculate the sum of the known angles: 88 + 118 + 132 + 100 = 438 .
Solving for x Now, we solve for x : x = 540 − 438 = 102 .
Final Answer Therefore, the measure of the fifth interior angle is 10 2 ∘ .
Examples
Understanding the angles in a pentagon can be useful in architecture. For example, if you're designing a pentagonal building, knowing the relationships between the interior angles helps ensure the structural integrity and aesthetic appeal of the design. By calculating the missing angle, architects can ensure that the building's corners align correctly and that the space is used efficiently. This principle applies not only to buildings but also to designing furniture or other objects with pentagonal shapes.
