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 degrees at the center is \(360^\circ\), and there are 5 congruent vertex angles. So each vertex angle \( \theta \) is \( \frac{360^\circ}{5} = 72^\circ \).

Step2: Calculate the base angle of the isosceles triangle

In an isosceles triangle, the sum of interior angles is \(180^\circ\). Let the base angle be \(x\). Since the triangle is isosceles, the two base angles are equal. So we have the equation \( 72^\circ + 2x = 180^\circ \). Solving for \(x\): \( 2x = 180^\circ - 72^\circ = 108^\circ \), then \( x = \frac{108^\circ}{2} = 54^\circ \). Wait, no, wait. Wait, no, the vertex angle is at the center, so the two equal angles are the base angles? Wait, no, the isosceles triangle has two equal sides (the radii) and the base is the side of the pentagon. So the vertex angle is at the center (between the two radii), and the two base angles are at the vertices of the pentagon. Wait, maybe I mixed up. Let's re - do.

The triangle is isosceles with two sides as the distance from the center to a vertex (let's call this length \(r\)). So the two equal angles are the base angles (at the vertices of the pentagon), and the vertex angle is at the center.

Sum of angles in a triangle: \(180^\circ\). Vertex angle (at center) is \( \frac{360^\circ}{5}=72^\circ\). Let the base angle be \(b\). Then \(2b + 72^\circ=180^\circ\). So \(2b=180 - 72 = 108^\circ\), so \(b = 54^\circ\)? Wait, but that's not right. Wait, no, maybe I got the triangle wrong. Wait, the regular pentagon's internal angle is \(108^\circ\), but that's the angle of the pentagon. Wait, no, the question is about the base angle of the isosceles triangle formed by the center and two vertices.

Wait, let's start over. The five triangles are congruent and isosceles, with a common vertex at the center. The angle at the center (vertex angle) for each triangle: since there are 5 triangles, the total around the center is \(360^\circ\), so each vertex angle is \(360\div5 = 72^\circ\). Now, in an isosceles triangle, the two base angles are equal. Let the base angle be \(x\). Then, using the angle - sum property of a triangle (\(180^\circ\)): \(72 + 2x=180\). Subtract \(72\) from both sides: \(2x = 180 - 72=108\). Then divide by 2: \(x = 54\)? Wait, but that contradicts my initial thought. Wait, no, maybe I made a mistake. Wait, no, the base angles of the isosceles triangle (the ones we are supposed to find) are equal. Wait, but let's check the answer options. The options are \(54^\circ\), \(72^\circ\), \(108^\circ\), \(144^\circ\). Wait, maybe I messed up the vertex angle. Wait, no, the vertex angle is \(72^\circ\), so the two base angles: \((180 - 72)/2 = 54\). But wait, another way: the internal angle of a regular pentagon is \(108^\circ\), which is the angle of the pentagon at each vertex. The triangle's base angle: wait, no, the triangle's base is the side of the pentagon, and the two equal sides are the radii. So the angle at the pentagon's vertex (the internal angle of the pentagon) is related to the triangle's angles. Wait, maybe I was wrong in step 2. Wait, no, let's recast.

Wait, the problem says "the measure of a base angle of one of the triangles". The triangle is isosceles, with vertex angle at the center (360/5 = 72 degrees). Then the other two angles (base angles) are equal. So (180 - 72)/2 = 54? But that's option A. Wait, but maybe I confused the base angle. Wait, no, the isosceles triangle has two equal sides (the radii), so the angles opposite those sides (the base angles) are equal. So the vertex angle is 72, so the base angles are (180 - 72)/2 = 54. So the answer should be 54 degrees.

Wait, but let's check again. Sum of angles in a triangle: 180. Vertex angle: 360/5 = 72. So 180 - 72 = 108, divided by 2 (since two base angles) is 54. Yes, that's correct.

Answer:

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