Sure! Let's solve this step-by-step:
1. Understanding the Problem:
An isosceles triangle has two equal sides and, consequently, two equal base angles. We're given that each base angle measures [tex]\(54^\circ\)[/tex].
2. Sum of Angles in a Triangle:
The sum of all interior angles in any triangle is [tex]\(180^\circ\)[/tex].
3. Setting Up the Equation:
Since it's an isosceles triangle, it has two base angles each measuring [tex]\(54^\circ\)[/tex]. Let's denote the base angles as [tex]\(A\)[/tex] and [tex]\(B\)[/tex], and the vertex angle as [tex]\(C\)[/tex].
Given: [tex]\(A = 54^\circ\)[/tex] and [tex]\(B = 54^\circ\)[/tex]
4. Calculating the Vertex Angle:
The sum of the angles in the triangle can be written as:
[tex]\[
A + B + C = 180^\circ
\][/tex]
Plugging in the known values for [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[
54^\circ + 54^\circ + C = 180^\circ
\][/tex]
5. Solving for [tex]\(C\)[/tex]:
Simplify the left-hand side of the equation:
[tex]\[
108^\circ + C = 180^\circ
\][/tex]
Subtract [tex]\(108^\circ\)[/tex] from both sides to solve for [tex]\(C\)[/tex]:
[tex]\[
C = 180^\circ - 108^\circ
\][/tex]
[tex]\[
C = 72^\circ
\][/tex]
6. Conclusion:
Therefore, the measure of the vertex angle [tex]\(C\)[/tex] is [tex]\(72^\circ\)[/tex].
The correct answer is [tex]\( \boxed{72^\circ} \)[/tex].
