Hello. This is problem 23 from your geometry textbook. This problem deals with angle by sectors. Yeah. When angles are bisected, they are divided into equal parts. Since I cannot draw the diagram, I've listed the main relationships that are evident from looking at the diagram. Um in your textbook, the overall outside angle, the largest angle is an angle B. A C, Angle BAC is broken up into four Interior angles by three race. We do know that the measure of this overall angle Is 120°. So when we add all four of these angles, it has to equal 120°. Now as we go through the problem and the information that they provide us, They take these four angles and Take the last two angles are created by ray A. D, bisecting angle B A. F. And the right to angles bisect are bisected well are created by ray E bisecting angle F. A C. So we're going to write some arbitrary information down here. So the measure of angle B a f is equal to the measure of angle B A. D plus the measure of angle D A F. We do not know what that exact angle measure is. All we know is that that angle B A F is bisected. So the measure of B a D is the same as a measure of D A F. So we're going to sign the value of X degrees to both of these. Then we'll do the same thing for the right two angles. The measure of angle F. A C. Secret to measure of angle F. E plus the measure of anger E A C nowhere in this problem do they indicate that ray A F divides um, angle be a bisects angle B A. C. So we are cannot be sure that the measure for B a F is the same as a measure for F A. C. But we do know that Ray E bisects angle F. A C. So we are going to assume that it gives it a separate, a different angle measure for both of those. So those are both B. Y degrees. Yeah. Now if both of those are wide agrees, then when we go through this angle um D. A. We want to find the measure of angle D. A. E. If we look at the information we have. So far, we've indicated that the measure of angle D A F. Is X degrees and the measure of angle F E. Is Y degrees. That still does not give us the exact number value yet for angle D A. E. Now we're going to go back and take our initial equation. They informed us that the overall measure of angle B a c Is 120°. If I combine that with the value of X degrees for B A. D and D. F. And why degrees for F A N E A C R equation now looks like this 120 is equal to X plus X plus why plus why That gives us that 120 Is equal to two x plus two Y. At first it looks like we're stuck. However, you should notice 120 and two are both even numbers. So I can factor out a factor of two from both of these And cancel that out. So 120 is two times 60. I factor out too. On this side I'm left with X plus Y. And now I can divide that factor of two from both sides. And it leads me to the equation that's 60 is equal to X plus Y. Where did I see that X plus Y Before this, X plus Y is the value for my angle D A. E. And now we know that X plus Y adds up to 60 degrees. We do not know the exact values for X and Y, but we do know that when they are added together, they give us 60. So the measure of anglo D E. Is 60 degrees.