Question

prove: (overline{ac} cong overline{de})
diagram with points a, b, c (horizontal), d, e (vertical through b) with congruence markings
statements

1. (overline{ab} cong overline{db}) and (overline{be} cong overline{bc}) \treasons
2. given
3. (\boldsymbol{\text{blank}}) \t2. def. of (cong) line segments
4. (ab + bc = ac), (db + be = de) \t3. segment addition postulate
5. (db + be = ac) \t4. (\boldsymbol{\text{blank}})
6. (ac = db + be) \t5. symmetric property of (=)
7. (\boldsymbol{\text{blank}}) \t6. transitive property of (=)
8. (overline{ac} cong overline{de}) \t7. (\boldsymbol{\text{blank}})

options:
a. (overline{ac} cong overline{de})
b. substitution
c. (ab = db), (be = bc)
d. def. of (cong) line segments
e. (ac = de)

Explanation:

Step1: Analyze Statement 2

The reason for statement 2 is "Def. of $\cong$ line segments". By the definition of congruent line segments, if $\overline{AB} \cong \overline{DB}$ and $\overline{BE} \cong \overline{BC}$, then their lengths are equal. So $AB = DB$ and $BE = BC$. So statement 2 should be option c: $AB = DB, BE = BC$.

Step2: Analyze Reason 4

Statement 4 is $DB + BE = AC$. We know from statement 2 that $AB = DB$ and $BE = BC$, and from statement 3, $AB + BC = AC$. So we substitute $AB$ with $DB$ and $BC$ with $BE$ in $AB + BC = AC$, which is the substitution property. So reason 4 is option b: Substitution.

Step3: Analyze Statement 6

Statement 5 is $AC = DB + BE$, and statement 3 is $DB + BE = DE$. By the transitive property of equality, if $AC = DB + BE$ and $DB + BE = DE$, then $AC = DE$. So statement 6 should be option e: $AC = DE$.

Step4: Analyze Reason 7

Statement 7 is $\overline{AC} \cong \overline{DE}$. We have $AC = DE$ from statement 6, and by the definition of congruent line segments (if two line segments have equal length, they are congruent), so reason 7 is option d: Def. of $\cong$ line segments.

Answer:

1. c. $AB = DB, BE = BC$
2. b. Substitution
3. e. $AC = DE$
4. d. Def. of $\cong$ line segments