Video Transcript
Find the measure of angle
𝐸𝐶𝐹 and the measure of angle 𝐴𝐵𝐹.
In this question, we’re asked
to find the measure of two angles, firstly the measure of angle 𝐸𝐶𝐹 and
secondly the measure of angle 𝐴𝐵𝐹. And in order to find these two
measures, we’ll use two properties of cyclic quadrilaterals. Firstly, we recall that
opposite angles in a cyclic quadrilateral sum to 180 degrees. And secondly, exterior angles
of a cyclic quadrilateral are equal to the interior angle at the opposite
vertex. Using the second property, we
see that the measure of angle 𝐸𝐶𝐹 is equal to the measure of the angle at
vertex 𝐴 and is therefore equal to 80 degrees.
Using the same property, the
measure of the exterior angle 𝐴𝐵𝐹 is equal to the measure of the interior
angle at vertex 𝐷, that is, the measure of angle 𝐴𝐷𝐶. Since angles on a straight line
sum to 180 degrees, we can calculate the measure of this angle by subtracting
104 degrees from 180 degrees. This is equal to 76
degrees. The measure of angle 𝐴𝐵𝐹 is
76 degrees. And we now have the two
solutions as required. Whilst we didn’t do so in this
question, we could have used the first property that opposite angles sum to 180
degrees to find the interior angles at vertices 𝐵 and 𝐶 first. We could then have used these
together with the fact that angles on a straight line sum to 180 degrees to find
the measures of 𝐸𝐶𝐹 and 𝐴𝐵𝐹. Either way, we end up with two
answers of 80 degrees and 76 degrees.
