Given that the measure of angle
𝐹𝐸𝐷 is equal to 14 degrees and the measure of angle 𝐶𝐵𝐴 is equal to two 𝑥
minus 96 degrees, calculate the value of 𝑥.
So let’s look at the diagram. We quickly see that arc 𝐴𝐶 is
congruent to arc 𝐷𝐹. And we know that inscribed angles
subtended by congruent arcs are going to be equal in measure. So this means that angle 𝐴𝐵𝐶 is
going to be equal in measure to angle 𝐷𝐸𝐹. We’re in fact told that the measure
of angle 𝐴𝐵𝐶 or 𝐶𝐵𝐴 is two 𝑥 minus 96. And the measure of angle 𝐹𝐸𝐷,
which is the same as 𝐷𝐸𝐹, is 14 degrees. Since these angles are equal, we
can form and solve an equation in 𝑥. That is, two 𝑥 minus 96 equals
14. To solve for 𝑥, we add 96 to both
sides, giving us two 𝑥 is equal to 110. And finally, we divide through by
two, and that gives us 𝑥 is equal to 55. And so given the information about
angles 𝐹𝐸𝐷 and 𝐶𝐵𝐴, we can deduce that 𝑥 is equal to 55.