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].
