Please give a step by step explanation with the reasons. Thank you.

Because angle 5 and angle 6 equal each other, it would mean that 4 and 7 equal each other. The sum of angles 4 and 5 is 180 because they are on the same line, intersected by another line. This means that they are supplementary angles. Since angles 5 and 6 equal each other, the other part of the the supplementary angle on both sides must equal each other. For example, if angles 5 and 6 equaled 60 degrees, the counterparts on the supplementary angle on both sides would have to be 120 to get to final sum of 180. This shows how angle 4 and 7 have to equal each other if 5 and 6 also equal other. Hope this helped! If its still confusing leave a comment and ill explain it in a different way. Good luck!