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Zorluk: Çok zorQuadrilaterals and Polygons

In a convex pentagon, two of the interior angles are equal in measure. The other three interior angles are also equal to each other, and each of these three is 40∘40^\circ larger than the measure of one of the first two angles. What is the measure of the largest exterior angle of this pentagon?

  1. A
    132∘132^\circ
  2. 96∘96^\circCevap
  3. C
    80∘80^\circ
  4. D
    48∘48^\circ

Cevap

The largest exterior angle is 96∘96^\circ.
The correct answer is 96∘96^\circ because setting up the equation for a pentagon's sum of interior angles, 2x+3(x+40)=5402x + 3(x + 40) = 540, gives x=84∘x = 84^\circ as the smallest interior angle. The exterior angle is supplementary to the interior angle, so 180∘−84∘=96∘180^\circ - 84^\circ = 96^\circ.

Adım Adım Çözüm

1
Determine the sum of the interior angles of a pentagon.
Using the formula (n−2)×180∘(n - 2) \times 180^\circ, for n=5n = 5, the sum is (5−2)×180∘=3×180∘=540∘(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
You must know the total sum of the interior angles before you can solve for any individual unknown angle.
2
Set up an algebraic equation to represent the interior angles.
Let the two smaller equal angles be xx. The three larger equal angles are (x+40∘)(x + 40^\circ). The equation is 2x+3(x+40∘)=540∘2x + 3(x + 40^\circ) = 540^\circ.
Translating the word problem into algebra allows us to solve for the specific angle measures.
3
Solve the equation for xx.
Distribute the 3 to get 2x+3x+120∘=540∘2x + 3x + 120^\circ = 540^\circ. Combine like terms: 5x+120∘=540∘5x + 120^\circ = 540^\circ. Subtract 120 from both sides: 5x=420∘5x = 420^\circ. Divide by 5: x=84∘x = 84^\circ.
Solving for xx provides the measure of the smallest interior angles.
4
Determine the largest exterior angle.
An exterior angle is supplementary to its adjacent interior angle. The largest exterior angle corresponds to the smallest interior angle. The largest exterior angle is 180∘−84∘=96∘180^\circ - 84^\circ = 96^\circ.
Because exterior and interior angles on a straight line add up to 180 degrees, the smallest interior angle creates the largest exterior angle.

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Quadrilaterals and Polygons
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