Four interior angles of a pentagon measure [tex]$88^{\circ}, 118^{\circ}, 132^{\circ}$[/tex], and [tex]$100^{\circ}$[/tex]. What is the measure of the fifth interior angle?
Answer (2)
The measure of the fifth interior angle of the pentagon is 10 2 ∘ . This is determined by calculating the total sum of the interior angles and subtracting the sum of the given angles from it. Thus, we find that 54 0 ∘ − 43 8 ∘ = 10 2 ∘ .
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Calculate the sum of the interior angles of a pentagon using the formula ( n − 2 ) × 18 0 ∘ , where n = 5 . The sum is 54 0 ∘ .
Sum the four given interior angles: 8 8 ∘ + 11 8 ∘ + 13 2 ∘ + 10 0 ∘ = 43 8 ∘ .
Subtract the sum of the four angles from the total sum of interior angles to find the fifth angle: 54 0 ∘ − 43 8 ∘ = 10 2 ∘ .
State the measure of the fifth interior angle: 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 of a Pentagon 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 ∘
Calculating the Sum of Known Angles Now, we solve for x . First, we calculate the sum of the known angles: 88 + 118 + 132 + 100 = 438
Solving for x Substitute this sum back into the equation: 43 8 ∘ + x = 54 0 ∘ Subtract 43 8 ∘ from both sides to find x : x = 54 0 ∘ − 43 8 ∘ = 10 2 ∘
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, when designing a building with a pentagonal base, architects need to ensure that the angles are correctly calculated to maintain structural integrity and aesthetic appeal. Knowing the sum of interior angles helps in precise planning and execution of the design.
