Question

1. complete the proof below by choosing the missing statements and justifications.

given: ( overline{ac} cong overline{df} ), ( overline{bc} cong overline{de} ). prove: ( overline{ab} cong overline{ef} )
(there is a diagram with segments: ( a---b---c ) and ( d---e---f ))

statements	justification
( overline{bc} cong overline{de} )	given

statements + justification bank:

• definition of congruence
• ( bc = de )
• ( ab + bc = ac )
• ( ac = df )
• segment addition postulate
• definition of congruence
• given
• given
• ( de + ef = df )
• ( ab cong ef )
• segment addition postulate
• ( ab cong ef )
• ( de + ef = df )
• substitution property of equality
• ( ab + bc = de + ef )
• transitive property
• ( ab + bc = de + ef )
• definition of congruence
• substitution property
• reflexive property
• symmetric property
• transitive property

Explanation:

Step1: Recall Segment Addition

We know from the segment addition postulate that for segment \( AC \), \( AB + BC = AC \), and for segment \( DF \), \( DE + EF = DF \).

Step2: Use Given Equalities

We are given \( AC = DF \) and \( BC = DE \). Substitute \( AC = AB + BC \) and \( DF = DE + EF \) into \( AC = DF \), we get \( AB + BC = DE + EF \).

Step3: Substitute \( BC = DE \)

Since \( BC = DE \), we can replace \( BC \) with \( DE \) in the equation \( AB + BC = DE + EF \), resulting in \( AB + DE = DE + EF \).

Step4: Subtract \( DE \) from Both Sides

Using the subtraction property of equality, subtract \( DE \) from both sides of \( AB + DE = DE + EF \). This gives \( AB = EF \).

Answer:

\( AB = EF \) (by using segment addition postulate, substitution with given equalities, and subtraction property of equality)