Question

complete the proof by choosing the correct reason.
given: bd = ce
prove: bc = de
1 bd = bc + cd
reason?
2 ce = cd + de
reason?
3 bd = ce
reason?
4 bc + cd = cd + de
reason?
5 bc = de
reason?

Explanation:

Step1: Given

$BD = BC + CD$ (Segment - addition postulate, which states that if a point $C$ lies on a line segment $\overline{BD}$, then $BD=BC + CD$)

Step2: Given

$CE=CD + DE$ (Segment - addition postulate, since point $D$ lies on line segment $\overline{CE}$)

Step3: Given

$BD = CE$ (Given in the problem statement)

Step4: Substitute

Since $BD = BC + CD$ and $CE=CD + DE$ and $BD = CE$, we can substitute: $BC + CD=CD + DE$ (Substitution property of equality)

Step5: Subtract $CD$ from both sides

$BC=DE$ (Subtraction property of equality: if $a + c=b + c$, then $a = b$)

Answer:

1. Reason: Segment - addition postulate
2. Reason: Segment - addition postulate
3. Reason: Given
4. Reason: Substitution property of equality
5. Reason: Subtraction property of equality