Question

complete the proof by choosing the correct
eason\.
given: ( cf = eg )
( df = fg )
prove: ( cd = ef )

statement	reason
2 ( eg = ef + fg )	reason
3 ( cf = eg )	given, reflexive property, transitive property, addition and subtraction properties, multiplication and division properties, substitution property, definition of congruent segments, segment addition property, definition of congruent angles, angle addition property
4 ( cd + df = ef + fg )	given, reflexive property, transitive property, addition and subtraction properties, multiplication and division properties, substitution property, definition of congruent segments, segment addition property, definition of congruent angles, angle addition property
5 ( df = fg )	given, reflexive property, transitive property, addition and subtraction properties, multiplication and division properties, substitution property, definition of congruent segments, segment addition property, definition of congruent angles, angle addition property
6 ( cd + fg = ef + fg )	given, reflexive property, transitive property, addition and subtraction properties, multiplication and division properties, substitution property, definition of congruent segments, segment addition property, definition of congruent angles, angle addition property
7 ( cd = ef )	given, reflexive property, transitive property, addition and subtraction properties, multiplication and division properties, substitution property, definition of congruent segments, segment addition property, definition of congruent angles, angle addition property

Explanation:

Step1: Match statement 1 to reason

The statement $CF = CD + DF$ states that a segment is equal to the sum of its two adjacent sub-segments, which fits the Segment Addition Property.

Step2: Match statement 2 to reason

The statement $EG = EF + FG$ follows the same logic as step 1, so it also uses the Segment Addition Property.

Step3: Match statement 3 to reason

The statement $CF = EG$ is explicitly provided as a given condition, so the reason is Given.

Step4: Match statement 4 to reason

Substitute $CF=CD+DF$ and $EG=EF+FG$ into $CF=EG$, so this uses the Substitution Property.

Step5: Match statement 5 to reason

The statement $DF = FG$ is explicitly provided as a given condition, so the reason is Given.

Step6: Match statement 6 to reason

Substitute $DF=FG$ into $CD + DF = EF + FG$, so this uses the Substitution Property.

Step7: Match statement 7 to reason

Subtract $FG$ from both sides of $CD + FG = EF + FG$, which uses the Addition and Subtraction Properties.

Answer:

1. Segment Addition Property
2. Segment Addition Property
3. Given
4. Substitution Property
5. Given
6. Substitution Property
7. Addition and Subtraction Properties