Short Answers (Explanation in next section below)
Reason A: Given
Reason B: Angle Addition Postulate
Reason C: Substitution Property
Reason D: Subtraction Property of Equality
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Further Explanation:
Reason A: At the very top of the problem, it states "given: ..." with the given angle property. All we do for the first line of any proof is repeat what is given to us. It seems silly and redundant, but it is important. Whenever you do a proof you always start with what you're given (some fact or theorem already proven earlier) and you somehow try to connect it to what you are trying to prove.
Reason B: The angle addition postulate is essentially the idea where we break up an angle into smaller pieces. This is probably better shown in a visual style instead of using words. Have a look at the attached image. Notice how the green angle ABD and the red angle DBE combine to form the purple angle ABE. Hopefully the image is clear enough. If not, then feel free to ask any questions.
Reason C: We're using the substitution property here. If x = a and x = b, then we can say that a = b. That's what is going on but on a more complicated level.
Reason D: We subtract the quantity "measure of angle DBE" from both sides. It is like going from x+2 = y+2 to x = y (subtract 2 from both sides). This is the subtraction property of equality in action. In general the rule is if a = b, then a-c = b-c
Reason A: Given
Reason B: Angle Addition Postulate
Reason C: Substitution Property
Reason D: Subtraction Property of Equality
----------------------------------------------------------
Further Explanation:
Reason A: At the very top of the problem, it states "given: ..." with the given angle property. All we do for the first line of any proof is repeat what is given to us. It seems silly and redundant, but it is important. Whenever you do a proof you always start with what you're given (some fact or theorem already proven earlier) and you somehow try to connect it to what you are trying to prove.
Reason B: The angle addition postulate is essentially the idea where we break up an angle into smaller pieces. This is probably better shown in a visual style instead of using words. Have a look at the attached image. Notice how the green angle ABD and the red angle DBE combine to form the purple angle ABE. Hopefully the image is clear enough. If not, then feel free to ask any questions.
Reason C: We're using the substitution property here. If x = a and x = b, then we can say that a = b. That's what is going on but on a more complicated level.
Reason D: We subtract the quantity "measure of angle DBE" from both sides. It is like going from x+2 = y+2 to x = y (subtract 2 from both sides). This is the subtraction property of equality in action. In general the rule is if a = b, then a-c = b-c