Question

1. complete the formal geometric proof:

given: ( b ) is the midpoint of ( overline{ac} ), and ( overline{ab} cong overline{de} ).
prove: ( overline{bc} cong overline{de} )
diagram: points ( a ), ( b ), ( c ) collinear with ( b ) between ( a ) and ( c ); points ( d ), ( e ) with ( overline{de} ) marked congruent to ( overline{ab} )
statements

1. ( b ) is the midpoint of ( overline{ac} ), and ( overline{ab} cong overline{de} )

reasons

1. given
2. (\text{dropdown})
3. def. of midpoint
4. (\text{dropdown})
5. transitive property of ( cong )

( \text{a. } overline{bc} cong overline{ab} )
( \text{b. } overline{bd} cong overline{be} )
( \text{c. } overline{bc} cong overline{de} )
( \text{d. } overline{ab} cong overline{ac} )

Explanation:

Step1: Analyze Step 2 (Def. of Midpoint)

The definition of a midpoint states that if \( B \) is the midpoint of \( \overline{AC} \), then \( \overline{AB} \cong \overline{BC} \). So the statement for step 2 should be \( \overline{BC} \cong \overline{AB} \) (option a).

Step2: Analyze Step 3 (Transitive Property of \( \cong \))

The transitive property of congruence says that if \( \overline{AB} \cong \overline{BC} \) (from step 2) and \( \overline{AB} \cong \overline{DE} \) (given), then \( \overline{BC} \cong \overline{DE} \). So the statement for step 3 should be \( \overline{BC} \cong \overline{DE} \) (option c).

Answer:

1. a. \( \overline{BC} \cong \overline{AB} \)
2. c. \( \overline{BC} \cong \overline{DE} \)