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 at the center

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

Step2: Use the triangle angle - sum property

Let the base angles of the isosceles triangle be \(x\). In a triangle, the sum of angles is \(180^{\circ}\). For an isosceles triangle with vertex angle \(v = 72^{\circ}\), we have \(x+x + v=180^{\circ}\) (by the angle - sum property of a triangle, \(A + B + C=180^{\circ}\), where \(A = B=x\) and \(C = v\)).
Substitute \(v = 72^{\circ}\) into the equation: \(2x+72^{\circ}=180^{\circ}\).
Solve for \(x\):

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Answer:

\(54^{\circ}\)