Question

a regular pentagon is created using the bases of five congruent isosceles triangles, joined at a common vertex. the total number of degrees in the center is 360°. if all five vertex angles meeting at the center are congruent, what is the measure of a base angle of one of the triangles?
○ 54°
○ 72°
○ 108°
○ 144°

Explanation:

Step1: Find the vertex angle of the isosceles triangle

The total angle at the center is \(360^\circ\), and there are 5 congruent triangles. So the vertex angle of each isosceles triangle is \(\frac{360^\circ}{5} = 72^\circ\).

Step2: Find the base angle of the isosceles triangle

In an isosceles triangle, the sum of interior angles is \(180^\circ\). Let the base angle be \(x\). Then \(2x + 72^\circ = 180^\circ\). Solving for \(x\): \(2x = 180^\circ - 72^\circ = 108^\circ\), so \(x = \frac{108^\circ}{2} = 54^\circ\)? Wait, no, wait. Wait, no, the vertex angle is at the center, so the two base angles are equal. Wait, no, I made a mistake. Wait, the vertex angle is \(72^\circ\), so the two base angles: \(180 - 72 = 108\), divided by 2 is \(54\)? But wait, no, wait the regular pentagon's central angle is \(72^\circ\), and the isosceles triangle has vertex angle \(72^\circ\), so base angles: \((180 - 72)/2 = 54\)? But the options have \(54^\circ\) as an option. Wait, but let me check again. Wait, the regular pentagon: when you divide it into 5 isosceles triangles from the center, each central angle (vertex angle of the triangle) is \(360/5 = 72^\circ\). Then, in the isosceles triangle, the two base angles are equal. So sum of angles in triangle is \(180\), so base angle is \((180 - 72)/2 = 54^\circ\). Wait, but the options: 54, 72, 108, 144. So 54 is an option. Wait, but maybe I messed up. Wait, no, the base angle: let's confirm. The regular pentagon's internal angle is \(108^\circ\), but that's the angle of the pentagon, not the triangle. The triangle's base angle: when you have the isosceles triangle with vertex angle \(72^\circ\), the base angles are \((180 - 72)/2 = 54^\circ\). So the answer should be \(54^\circ\). Wait, but let me check again. Wait, the problem says "a base angle of one of the triangles". So yes, the vertex angle is \(72^\circ\), so base angles are \((180 - 72)/2 = 54^\circ\). So the correct option is \(54^\circ\).

Answer:

\(54^\circ\) (corresponding to the option with \(54^\circ\))