a Let's draw the regular pentagon ABCDE inscribed in circle ⊙ O.
b The sum of a polygon's interior angles can be calculated using the formula 180^(∘)(n-2), where n is the number of sides. Since the interior angles of a regular polygon are all congruent, we can find the measures of any one of them by dividing the formula for finding the sum of the interior angles by the number of sides, n.
To find the measure of ∠ BOC we can add a few more segments, creating five congruent triangles. We know they are congruent because they have 3 pairs of congruent sides.
The 5 vertex angles of these triangles sum to 360^(∘). Therefore, by dividing 360^(∘) by 5, we get the measure of ∠ BOC.
m∠ BOC=360^(∘)/5= 72^(∘)
d mEBC refers to the measure of the intercepted arc from E to C that also passes through B. Let's highlight that arc in our diagram.
The central angle that spans mEBC equals the sum of three of the vertex angles that we identified in Part C. Since one of these vertex angles is 72^(∘), three times this must mean that the central angle is 72^(∘)(3)=216^(∘). The intercepted arc will have the same measure as the central angle.
Therefore, we know that mEBC = 216^(∘).
Is There Another Way?
We could also find the inscribed angle to the intercepted arc EBC which is ∠ EDC. This angle coincide with one of the interior angles of ABCDE which we know from Part B is 108^(∘). Let's add this to the diagram.
The inscribed angle, ∠ EDC, is always twice the measure of its corresponding central angle ∠ EOC. Since the inscribed angle is 108^(∘) the central angle must be 216^(∘), which is also the measure of the intercepted arc.
