Question

1. given: \\(\overline{ab} \cong \overline{cd}\\), \\(\overline{bd} \cong \overline{de}\\)

prove: \\(\overline{ad} \cong \overline{ce}\\)
\

$$\begin{tabular}{|l|l|} \\hline \\textbf{statements} & \\textbf{reasons} \\\\ \\hline 1. \\(\\overline{ab} \\cong \\overline{cd}\\), \\(\\overline{bd} \\cong \\overline{de}\\) & 1. given \\\\ \\hline 2. & 2. definition of congruence \\\\ \\hline 3. \\(ab + bd = ad\\) & 3. \\\\ \\hline 4. \\(cd + de = ad\\) & 4. \\\\ \\hline 5. & 5. segment addition postulate \\\\ \\hline 6. \\(ad = ce\\) & 6. \\\\ \\hline 7. & 7. definition of congruence \\\\ \\hline \\end{tabular}$$

(image: a line segment ad with b between a and d, and a perpendicular segment cde with d between c and e)

Explanation:

Step 1: Analyze Statement 2

The definition of congruence for segments states that if two segments are congruent, their lengths are equal. So from \(\overline{AB} \cong \overline{CD}\) and \(\overline{BD} \cong \overline{DE}\), we get \(AB = CD\) and \(BD=DE\).
Statement 2: \(AB = CD\), \(BD = DE\)

Step 2: Analyze Statement 3's Reason

The reason for \(AB + BD=AD\) is the Segment Addition Postulate, which states that if \(B\) is between \(A\) and \(D\), then \(AB + BD=AD\).
Reason 3: Segment Addition Postulate

Step 3: Analyze Statement 4's Reason

We know \(AB = CD\) (from step 2) and \(BD = DE\) (from step 2). Substitute \(AB\) with \(CD\) and \(BD\) with \(DE\) in \(AB + BD=AD\). So by Substitution Property, \(CD + DE=AD\).
Reason 4: Substitution Property (substituting \(AB = CD\) and \(BD = DE\) into \(AB + BD = AD\))

Step 4: Analyze Statement 5

The Segment Addition Postulate for \(C\), \(D\), \(E\): if \(D\) is between \(C\) and \(E\), then \(CD + DE=CE\).
Statement 5: \(CD + DE=CE\)

Step 5: Analyze Statement 6's Reason

From statement 4 (\(CD + DE = AD\)) and statement 5 (\(CD + DE=CE\)), by Transitive Property of Equality (if \(a = b\) and \(b = c\), then \(a = c\)), we get \(AD=CE\).
Reason 6: Transitive Property of Equality (or Substitution, since \(CD + DE\) is equal to both \(AD\) and \(CE\))

Step 6: Analyze Statement 7

The definition of congruence says that if two segments have equal length, they are congruent. Since \(AD = CE\), then \(\overline{AD} \cong \overline{CE}\).
Statement 7: \(\overline{AD} \cong \overline{CE}\)

Filling the table:

Statements	Reasons
2. \(AB = CD\), \(BD = DE\)	2. Definition of Congruence
3. \(AB + BD=AD\)	3. Segment Addition Postulate
4. \(CD + DE = AD\)	4. Substitution Property (substituting \(AB = CD\) and \(BD = DE\) into \(AB + BD=AD\))
5. \(CD + DE=CE\)	5. Segment Addition Postulate
6. \(AD = CE\)	6. Transitive Property of Equality (or Substitution)
7. \(\overline{AD} \cong \overline{CE}\)	7. Definition of Congruence

Answer:

Filled table as above. The key steps are using the definition of congruence to convert congruent segments to equal lengths, applying the Segment Addition Postulate, substitution, and then converting back to congruent segments. The final answer for the proof is that \(\overline{AD} \cong \overline{CE}\) is proven through the above - mentioned logical steps.