Question

fill in the blank 10 points

1. given: (overline{ab} cong overline{cd})

prove: (overline{ac} cong overline{bd}) (\boldsymbol{a}) (\boldsymbol{b}) (\boldsymbol{c}) (\boldsymbol{d})

statements	reasons
2.) (ab = cd)	2) choose your answer...
3.) (ac +) choose your answer... (= ad)	3) choose your answer...
4.) (ab + bd = ad)	4) choose your answer...
5.) (cd + bd = ad)	5) choose your answer...
6.) (ac + cd = cd + bd)	6) choose your answer...
7.) (ac = bd)	7) choose your answer...
8.) (overline{ac} cong overline{bd})	8) choose your answer...

Explanation:

Step 1: Identify Given

We know that \(\overline{AB}\cong\overline{CD}\) (given). So statement 1 is \(\overline{AB}\cong\overline{CD}\) and reason 1 is "Given".

Step 2: Congruent to Equal

If two segments are congruent, their lengths are equal. So \(AB = CD\) (reason: Definition of congruent segments).

Step 3: Segment Addition Postulate (AC + CD = AD)

By the Segment Addition Postulate, for points \(A\), \(C\), \(D\) on a line, \(AC + CD=AD\). So the missing segment in step 3 is \(CD\), and reason 3 is "Segment Addition Postulate".

Step 4: Segment Addition Postulate (AB + BD = AD)

By the Segment Addition Postulate, for points \(A\), \(B\), \(D\) on a line, \(AB + BD = AD\). Reason 4 is "Segment Addition Postulate".

Step 5: Substitution (CD + BD = AD)

Since \(AB = CD\) (from step 2), we substitute \(AB\) with \(CD\) in \(AB + BD = AD\) to get \(CD + BD = AD\). Reason 5 is "Substitution Property".

Step 6: Substitution (AC + CD = CD + BD)

From step 3 (\(AC + CD = AD\)) and step 5 (\(CD + BD = AD\)), we can say \(AC + CD=CD + BD\) (since both equal \(AD\)). Reason 6 is "Substitution Property (or Transitive Property of Equality)".

Step 7: Subtraction Property (AC = BD)

Subtract \(CD\) from both sides of \(AC + CD = CD + BD\), we get \(AC = BD\). Reason 7 is "Subtraction Property of Equality".

Step 8: Equal to Congruent (AC ≅ BD)

If two segments have equal length, they are congruent. So \(\overline{AC}\cong\overline{BD}\) (reason: Definition of congruent segments).

Filling the Table:

Statements	Reasons
2. \(AB = CD\)	2. Definition of congruent segments
3. \(AC + \boldsymbol{CD}=AD\)	3. Segment Addition Postulate
4. \(AB + BD = AD\)	4. Segment Addition Postulate
5. \(CD + BD = AD\)	5. Substitution Property (from \(AB = CD\))
6. \(AC + CD = CD + BD\)	6. Substitution Property (both equal \(AD\))
7. \(AC = BD\)	7. Subtraction Property of Equality (subtract \(CD\))
8. \(\boldsymbol{\overline{AC}\cong\overline{BD}}\)	8. Definition of congruent segments

Answer:

The completed proof has the statements and reasons as filled in the table above. The key steps involve using the given congruence, segment addition postulate, substitution, and properties of equality to show the required congruence.