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Given: line ae is congruent to line bd, line cd is congruent to line ce prove: line ac is congruent to line bc

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1 Answer

First, generally, "lines" cannot be described as congruent because they have no definite length, so I think you are describing line segments.  Line segments are congruent if their lengths are equal. 
Second, I think there is information in your picture that is not being conveyed in the text of the question.  I suspect that your picture might look something like this
A          B
  \        /
   \      /
    \    /
     \  /
     / \
    /   \
   /     \
  /       \
D         E
So a two line proof of this using this figure might look something like this
1. AE ≅ BD                   Given
2. CD ≅ CE                   Given
3. AE = BD                   Def. of congruent segments
4. CD = CE                   Def. of congruent segments.
5. AE = AC + CE           Whole is equal to the sum of its parts
6. BD = BC + CD          Whole is equal to the sum of its parts
7. AC + CE = BC + CD   Substitution lines 3, 5 & 6
8. AC + CD = BC + CD   Substitution lines 4 & 7
9. AC = BC                    Subtracting CD from both sides of line 8 (Equals subtracted by equals are equal).
10. AC ≅ BC                  Def. of congruent segments.
The key points to take away from such a proof are to (1) understand relationship between congruence and equality of measure and (2) unpack information from definitions and postulates. Stay warm, and I hope this was helpful.  John

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