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What is the measure of each interior angle of a regular deca

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by Max@Math Revolution » Tue Jan 22, 2019 11:54 pm

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[Math Revolution GMAT math practice question]

What is the measure of each interior angle of a regular decagon?

A. 72°
B. 108°
C. 120°
D. 135°
E. 144°
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by fskilnik@GMATH » Wed Jan 23, 2019 3:24 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

What is the measure of each interior angle of a regular decagon?

A. 72°
B. 108°
C. 120°
D. 135°
E. 144°
$$\sum\nolimits_N {\,\, = \,\,\,\left( {N - 2} \right) \cdot 180\,\,\,\,\,\, \Rightarrow \,\,\,\,\,} \sum\nolimits_{10} {\,\, = \,\,\,\left( {10 - 2} \right) \cdot 180 = 8 \cdot 180} \,\,\,\,\,\,\,\left[ {{\text{degrees}}} \right]$$
$$?\,\,\,\mathop = \limits^{{\text{regular}}} \,\,\,\frac{{\sum\nolimits_{10} {} }}{{10}} = \underleftrightarrow {8 \cdot \left( {20 - 2} \right) = 160 - 16} = 144\,\,\,\left[ {{\text{degrees}}} \right]$$

We follow the notations and rationale taught in the GMATH method.

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by Brent@GMATPrepNow » Wed Jan 23, 2019 5:29 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

What is the measure of each interior angle of a regular decagon?

A. 72°
B. 108°
C. 120°
D. 135°
E. 144°
Useful rule: the sum of the angles in an n-sided polygon = (n - 2)(180º)
So, for example, the sum of the angles in an 11-sided polygon = (11 - 2)(180º)
= (9)(180º)
= 1620º

What is the measure of each interior angle of a regular decagon?
A decagon has 10 sides.
The sum of the angles in an 10-sided polygon = (10 - 2)(180º)
= (8)(180º)
= 1440º

In a REGULAR polygon, all angles are equal
So, in a decagon, each angle = 1440º/10 = 144º

Answer: E

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by Max@Math Revolution » Fri Jan 25, 2019 1:05 am
=>

The sum of all interior angles of n-gon is (n-2)*180°.
A decagon is a 10-gon.
The sum of all interior angles is (10-2)* 180° = 8*180°.
And each interior angle of a regular decagon has measure 8*180° / 10 = 8*18° = 144°.

Therefore, the answer is E.
Answer: E
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by Scott@TargetTestPrep » Sun Jan 27, 2019 6:12 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

What is the measure of each interior angle of a regular decagon?

A. 72°
B. 108°
C. 120°
D. 135°
E. 144°
A decagon is a polygon with 10 sides and thus 10 angles. We use the formula for the total number of degrees in an n-gon as n(180 - 2).

Thus, the sum of the measures of all 10 angles is 180(10 - 2) = 180(8) = 1440 degrees. A regular decagon is one that has all equal sides and equal angles. Thus, the measure of each angle of a regular decagon is 1440/10 = 144 degrees.

Alternate Solution:

We can also use the fact that the exterior angles of all polygons add up to 360 degrees. Since a regular decagon has all equal sides and equal interior angles, all the exterior angles are also equal and thus, an exterior angle of a regular decagon is 360/10 = 36 degrees. If an exterior angle is 36 degrees, then an interior angle is 180 - 36 = 144 degrees.

Answer: E

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