Answers:
Reason 4: Isosceles triangle
Reason 5: Segment Addition Postulate
Reason 6: Segment Addition Postulate
Reason 7: Substitution
Reason 8: Substitution
Reason 9: Subtraction property of equality
Reason 10: Definition of congruence
Reason 11: Vertical Angles Theorem
Reason 12: Vertical Angles Theorem
Reason 13: ASA (angle side angle) congruence theorem
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Explanations:
Reason 4: Triangle DCE is an isosceles triangle because of statement 1 of the proof table. The congruent base angles lead to congruent opposite sides.
Reason 5 and 6: This is the idea of breaking any segment into smaller pieces. The smaller pieces glue back together to get the larger original segment.
Reasons 7 and 8: Substitution is a math term which means "replacement". So for instance, in line 7 we started with DB = DC + CB (from line 6), then we replaced the "DB" with "DF" (see line 3); and then replaced "DF" with "DE+EF"
Reason 9: In this step, we subtract DE from both sides to have it go away and cancel out
Reason 10: If two lengths are equal, then the segments are congruent
Reason 11 & 12: Two angles are vertical if they are opposite from one another after forming from two lines crossing. Think of an X shape. Vertical angles are congruent.
Reason 13: We have two pairs of congruent angles, and a pair of congruent sides. The sides are between the angles. The order is important. See the attached image below.