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: Calculate vertex - angle at center

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

Step2: Use angle - sum property of triangle

Let the base - angle of the isosceles triangle be $x$. In an isosceles triangle, the two base - angles are equal. By the angle - sum property of a triangle ($180^{\circ}$ for the sum of interior angles), we have $x + x+72^{\circ}=180^{\circ}$, which simplifies to $2x=180^{\circ}-72^{\circ}=108^{\circ}$.

Step3: Solve for base - angle

Dividing both sides of $2x = 108^{\circ}$ by 2, we get $x = 54^{\circ}$.

Answer:

$54^{\circ}$