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? 108° 144° 54° 72°

Explanation:

Step1: Find vertex angle of triangle

The total degrees 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: Calculate base angle of triangle

In an isosceles triangle, the sum of angles is \(180^\circ\). Let the base angle be \(x\). Then \(2x + 72^\circ=180^\circ\). Solving for \(x\), we get \(2x = 180^\circ - 72^\circ = 108^\circ\), so \(x=\frac{108^\circ}{2}=54^\circ\)? Wait, no, wait. Wait, the vertex angle at the center is \(72^\circ\), so the two base angles (the ones we need) are equal. So \(180 - 72 = 108\), divided by 2 is \(54\)? Wait, no, maybe I misread. Wait, the question is "what is the measure of a base angle of one of the triangles". Wait, the vertex angle is \(72^\circ\) (since \(360/5 = 72\)). Then the base angles: in an isosceles triangle, base angles are equal. So sum of angles is \(180\), so \(180 - 72 = 108\), divided by 2 is \(54\)? Wait, but let's check again. Wait, the regular pentagon is made by five congruent isosceles triangles joined at the center. The central angle (vertex angle of each triangle) is \(360/5 = 72^\circ\). Then the base angles: each triangle has two equal sides (the radii), so it's isosceles with vertex angle \(72^\circ\). So base angles: \((180 - 72)/2 = 54^\circ\)? Wait, but maybe I made a mistake. Wait, no, let's recalculate. \(360\) divided by \(5\) is \(72\), so the vertex angle is \(72\). Then the other two angles: \(180 - 72 = 108\), divided by \(2\) is \(54\). So the base angle is \(54^\circ\)? Wait, but let's check the options. The options are \(108^\circ\), \(54^\circ\), \(72^\circ\), \(144^\circ\). Wait, maybe I messed up the vertex angle. Wait, no, the total at the center is \(360\), five triangles, so each central angle is \(72\). Then the base angles: in the isosceles triangle, the two equal angles are the base angles. So \(180 - 72 = 108\), divided by 2 is \(54\). So the base angle is \(54^\circ\)? Wait, but let's confirm. Alternatively, maybe the vertex angle is \(108\)? No, no, the central angle is \(360/5 = 72\). So the triangle has vertex angle \(72\), so base angles are \(54\) each. So the answer should be \(54^\circ\)? Wait, but let's check again. Wait, the regular pentagon's internal angle is \(108^\circ\), but that's the angle of the pentagon, not the triangle. Wait, the triangles are isosceles with two sides as radii (equal) and the base as the side of the pentagon. So the central angle (between the two radii) is \(72^\circ\), so the base angles (at the base, which is the side of the pentagon) are \((180 - 72)/2 = 54^\circ\). Yes, that makes sense. So the base angle is \(54^\circ\). Wait, but let's check the options. The options are \(108\), \(54\), \(72\), \(144\). So \(54^\circ\) is an option. So that's the answer.

Answer:

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