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A regular pentagon and regular hexagon with sides the same l

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A regular pentagon and regular hexagon with sides the same l

by Max@Math Revolution » Thu Nov 07, 2019 11:11 pm
[GMAT math practice question]

Image

A regular pentagon and regular hexagon with sides the same length of sides are pasted as the figure shows. What is ∠ x+∠ y+∠ z+∠ w?

A. 283
B. 320
C. 324
D. 327
E. 331
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by deloitte247 » Sun Nov 10, 2019 3:47 am
A regular polygon with equal length has equal (congruent) angles.
The sum of all the 5 interior angles in a regular pentagon is 540 degrees.
The sum of all the 6 interior angles in a regular pentagon is 720 degrees.
Angle on a straight line is 180 degrees
$$Each\ angle\ in\ a\ regular\ pentagon=\frac{540^0}{5}=108^0$$
$$Each\ angle\ in\ a\ regular\ hexagon=\frac{720^0}{6}=120^0$$
$$Therefore,\ \angle x^0=180^0-120^0=60^0$$
$$\angle w^0=180^0-108^0=72^0$$
$$\angle z^0=180^0-120^0=60^0$$
$$\angle w^0+x^0+y^0+z^0=72^0+60^0+132^0+60^0=324^0$$
Answer = option C
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by Max@Math Revolution » Sun Nov 10, 2019 6:49 pm
=>

The interior angle of a regular hexagon is 180(6-2)/6 = 120, and the interior angle of a regular pentagon is 180(5-2)/5 = 108.
Thus ∠ x = ∠ z = 180 - 120 = 60 and ∠w = 180 - 108 = 72 and ∠ y = 360 - 120 - 108 = 132.
Thus ∠ x + ∠ y + ∠ z + ∠ w = 60 + 132 + 60 + 72 = 324.

Therefore, C is the answer.
Answer: C
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