We have been told that all the arcs in ⊙ P have integer measures.
Using this information, we will prove that x must be even. We will begin by identifying each arc measure. Since ∠ APB is the central angle of AB, the measure of AB is also x^(∘).
mAB=x^(∘)
In the same circle or congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
By this theorem, we can conclude that AC and BC are congruent. Remember that congruent arcs have the same measure, so let's suppose they measure y ^(∘).
mAC=mBC= y^(∘)
Knowing that the measure of the entire circle is 360^(∘), we will write x in terms of y.
x+ y+ y=360 ⇔ x=360- 2y
Note that any integer that can be divided by 2 is an even number. Therefore, both 360 and 2y are even numbers.
x=360_(even)-2y_(even)
Since subtracting an even number from an even number always results in an even number, x must also be an even number.
