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If four angles of a pentagon measure [tex]$108^{\circ}$[/tex], [tex]$115^{\circ}$[/tex], [tex][tex]$89^{\circ}$[/tex][/tex], and [tex]$102^{\circ}$[/tex], find the measure of the fifth angle.

A. [tex]$126^{\circ}$[/tex]
B. [tex][tex]$54^{\circ}$[/tex][/tex]
C. [tex]$306^{\circ}$[/tex]
D. [tex]$540^{\circ}$[/tex]
E. None of the other answers are correct



Answer :

GinnyAnswer
To find the measure of the fifth angle of a pentagon when four angles are given, we can use the properties of the interior angles of a pentagon.

1. Sum of Interior Angles of a Pentagon:
The sum of the interior angles of a pentagon can be calculated using the formula for the sum of interior angles of an [tex]\(n\)[/tex]-sided polygon:
[tex]\[ \text{Sum of interior angles} = (n-2) \times 180^\circ \][/tex]
For a pentagon ([tex]\(n = 5\)[/tex]):
[tex]\[ \text{Sum of interior angles} = (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \][/tex]

2. Sum of the Known Angles:
Add the four given angles:
[tex]\[ 108^\circ + 115^\circ + 89^\circ + 102^\circ = 414^\circ \][/tex]

3. Calculate the Fifth Angle:
Subtract the sum of the four known angles from the total sum of the interior angles of the pentagon:
[tex]\[ \text{Fifth angle} = 540^\circ - 414^\circ = 126^\circ \][/tex]

Therefore, the measure of the fifth angle is:
[tex]\[ \boxed{126^\circ} \][/tex]

Out of the given choices:
- [tex]\(126^{\circ}\)[/tex]
- [tex]\(54^{\circ}\)[/tex]
- [tex]\(306^{\circ}\)[/tex]
- [tex]\(540^{\circ}\)[/tex]
- None of the other answers are correct

The correct answer is [tex]\(126^{\circ}\)[/tex].

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