Given that the measure of angle
𝐴𝐸𝐶 equals 36 degrees and the measure of angle 𝐵𝐴𝐷 equals four 𝑥 minus eight
degrees, determine the value of 𝑥.
We are given, firstly, that the
measure of angle 𝐴𝐸𝐶 is 36 degrees. The measure of angle 𝐵𝐴𝐷 is four
𝑥 minus eight degrees. We don’t have any theorems that
tell us immediately how these two angles are related. So let’s observe that we are given
a set of parallel line segments. The line segment 𝐴𝐵 is parallel
to the line segment 𝐶𝐷. And so we can say that this angle
of 𝐴𝐷𝐶 must be alternate to angle 𝐵𝐴𝐷. So they will have equal angle
measures of four 𝑥 minus eight degrees.
Now, we can use the property that
inscribed angles subtended by the same arc are equal. We should observe that this arc
𝐴𝐶 subtends two angles, angle 𝐴𝐷𝐶 and angle 𝐴𝐸𝐶, meaning that these two
angle measures will be the same. And so we can set up an equation
that four 𝑥 minus eight degrees is equal to 36 degrees. Since the brackets on the left-hand
side are simply there just to indicate that this is a measure in angles, then we can
drop the brackets and the degrees from both sides of the equation. We can then add eight to both
sides, giving us four 𝑥 is equal to 44. Dividing both sides of the
equation, we have the answer that 𝑥 is equal to 11.