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° 144° 108° 72°

Explanation:

Step1: Find the vertex angle at the center

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

Step2: Use the angle - sum property of a triangle

Let the base angles of the isosceles triangle be \(x\). For an isosceles triangle, the sum of interior angles is \(180^{\circ}\). We know that in an isosceles triangle with vertex angle \(72^{\circ}\), \(x + x+72^{\circ}=180^{\circ}\) (angle - sum property of a triangle: \(A + B + C=180^{\circ}\), where \(A = B\) for an isosceles triangle)

$$2x=180^{\circ}- 72^{\circ}$$

$$2x = 108^{\circ}$$

$$x=\frac{108^{\circ}}{2}=54^{\circ}$$

Answer:

\(54^{\circ}\)