Video Transcript
𝐴𝐵𝐶𝐷𝐸 is the regular pentagon drawn inside the circle 𝑀. The line 𝐴𝑋 is a tangent to the circle at 𝐴, and the line 𝐸𝑋 is a tangent to the circle at 𝐸. Find the measure of angle 𝐴𝑋𝐸.
Let’s look carefully at the diagram we’ve been given. We can see that angle 𝐴𝑋𝐸 is the angle formed by two tangents, the lines 𝐴𝑋 and 𝐸𝑋, which intersect outside a circle. We need to recall the angles of intersecting tangents theorem. This tells us that the measure of the angle between two tangents that intersect outside a circle is half the positive difference of the measures of the intercepted arcs. The minor arc intercepted by these two tangents is the arc 𝐴𝐸. And the major intercepted arc is the arc which we can refer to as 𝐴𝐵𝐸, as 𝐵 is a point on this arc. So, by the angles of intersecting tangents theorem, the measure of angle 𝐴𝑋𝐸 is equal to a half the measure of the arc 𝐴𝐵𝐸 minus the measure of the arc 𝐴𝐸.
Now, we haven’t been given the measures of any angles or any arcs in this figure. The only other information we’ve got is that this pentagon is regular. This means that it can be divided into five congruent triangles by drawing in the radii from each vertex of the pentagon to the center of the circle. But how does this help us? Well, we know that the measure of an arc is defined to be equal to the measure of its central angle. The measure of the minor arc 𝐴𝐸 is equal to the angle between the two radii 𝐴𝑀 and 𝐸𝑀, as these connect the endpoints of this arc to the center of the circle. In the same way, the measure of the major arc 𝐴𝐵𝐸 is equal to the reflex angle at the center of the circle.
As the pentagon is regular, and so the five triangles are congruent, we know that the five angles at the center of the circle are the same. And so as angles around a point sum to 360 degrees, the measure of each of these angles is one-fifth of 360 degrees. That’s 72 degrees. So if the measure of angle 𝐴𝑀𝐸 is 72 degrees, then the measure of the arc 𝐴𝐸 is also 72 degrees. In the same way, if the measure of the reflex angle 𝐴𝑀𝐸 is four multiplied by 72 degrees, which is 288 degrees, then this is the measure of the major arc 𝐴𝐵𝐸. So we do in fact know the measures of both arcs. And substituting them into the equation we wrote using the angles of intersecting tangents theorem, we have that the measure of angle 𝐴𝑋𝐸 is a half of 288 degrees minus 72 degrees. That’s a half of 216 degrees, which is 108 degrees.
So by recalling the angles of intersecting tangents theorem and the fact that the measure of an arc is defined to be equal to the measure of its central angle, we found that the measure of angle 𝐴𝑋𝐸 is 108 degrees.
