Given that the measure of angle 𝐶𝐴𝐵 equals 39 degrees and 𝐸 is the midpoint of line segment 𝐴𝐶, find the measure of angle 𝐵𝐹𝐸.
So let’s have a look at this figure that we’re given, and we can fill in the measure of angle 𝐶𝐴𝐵, which is 39 degrees. The angle measure that we wish to find out is the measure of angle 𝐵𝐹𝐸, which is here. The center of the circle is here at point 𝑀, and we have two chords in the circle. The line segment 𝐴𝐵 is a chord and so is the line segment 𝐴𝐶. They are chords because both of these lines is a line segment joining two distinct points on the circumference.
Taking a closer look at the chord 𝐴𝐶, we can see that it’s divided into two equal or congruent pieces. In fact, we could also say that this chord 𝐴𝐶 has been bisected.
Let’s recall that there is an important property which involves a line from the center of a circle to a chord that has been bisected. This property tells us that if we have a circle with center 𝐴 containing a chord, line segment 𝐵𝐶, then the straight line that passes through 𝐴 and bisects line segment 𝐵𝐶 is perpendicular to line segment 𝐵𝐶. We can even say by changing the letters in the general statement that this is exactly what we have here. The chord 𝐴𝐶 has been bisected by this line from the center 𝑀. And so that means that this line is perpendicular to the chord.
Let’s consider how this piece of information is useful. Well, we can take this triangle 𝐴𝐹𝐸 and consider that we have two angles that we know and one which we don’t know. In fact, if we did know this angle measure of 𝐴𝐹𝐸, then we could work out the measure of the angle 𝐵𝐹𝐸 that we’re asked to calculate. We can use the property that the interior angle measures in a triangle add to 180 degrees to help us work out the angle measure of 𝐴𝐹𝐸.
So we can say that 90 degrees plus 39 degrees plus the measure of angle 𝐴𝐹𝐸 must be equal to 180 degrees. We can simplify 90 degrees plus 39 degrees as 129 degrees and then subtract this value from both sides of the equation, giving us that the measure of angle 𝐴𝐹𝐸 is 51 degrees. Next, we can work out the required angle measure of angle 𝐵𝐹𝐸. And we can do that by remembering that the angles on a straight line add up to 180 degrees.
That means that 51 degrees plus the measure of angle 𝐵𝐹𝐸 will be equal to 180 degrees. This means, of course, that the measure of angle 𝐵𝐹𝐸 can be found by subtracting 51 degrees from both sides, giving us an answer of 129 degrees. And so we find the answer using this very important property, which allowed us to work out that the line which bisected the chord and passes through the center is also perpendicular to the chord.