a Let's take a look at the given diagram of the pentagram.
We want to find the value of a using the fact that the pentagram can be inscribed in a circle. We know that angles at all of the five outer vertices of a pentagram are congruent to each other. Now, let's notice that the pentagram divides the circumference of the circle into 5 congruent arcs.
Since the measure of the arc going around the entirety of a circle is 360^(∘), the measure of each of the smaller arcs is 360^(∘) / 5 = 72^(∘). Now, a is the measure of the inscribed angle that has one of the smaller arcs as its intercepted arc. By the Inscribed Angle Theorem the value of a is half that of the intercepted arc.
a = 72^(∘)/2 = 36^(∘)
b We want to find the values of b, c, and d from the following diagram.
Let's start by finding the value of b.
b
In Part A, we found that a = 36^(∘). Also, since all angles at the outer vertices of the pentagram are congruent, all of them have a measure of 36^(∘).
Now, notice that we can find the measure of the reflexive angle adjacent to the b angle. The reflexive angle is the only missing angle of a concave quadrilateral.
Let's call the measure of the reflex angle x. In a quadrilateral, the sum of measures of interior angles equals 360^(∘). Therefore, we can write the following equation for x.
x + 36^(∘)+36^(∘)+36^(∘) = 360^(∘)
Let's solve it!
Since b and x are adjacent angles forming a complete angle, the sum of their measures equals 360^(∘). Knowing that x = 252^(∘), we can write the following equation for b.
b + 252^(∘) = 360^(∘)
⇒
b = 108^(∘)
We have that b = 108^(∘). Now, let's find the measure of c.
c
We know that b = 108^(∘). Since vertical angles have equal measures, the angle opposite the b angle also has the measure of 108^(∘).
Now, since all angles at the outer vertices of the pentagram are congruent, we have a regular pentagon within the pentagram. All interior angles of this pentagon are congruent. We conclude that c = 108^(∘).
Finally, let's find the measure of d.
d
Let's notice that d is the measure of an angle adjacent to one of the interior angles of the inner pentagon. These two angles form a straight line so the sum of their measures equals 180^(∘).
d + 108^(∘) = 180^(∘)
⇒
d = 72^(∘)
