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Answer :
We have proven that JK ≅ LM by showing that JL ≅ KM, using the segment addition postulate, and using the definition of congruence.
The completed proof:
Statements Reasons
a. JL ≅ KM Given
b. JL = KM Definition of congruence
c. JK + KL = JL; KL + LM = KM Segment Addition Postulate
d. JK + KL = KL + LM Substitution
e. JK + KL - KL = KL + LM - KL Subtraction Property of Equality
f. JK = LM Substitution
g. JK ≅ LM Definition of congruence
- To prove that JK ≅ LM, we can use the following steps:
- Show that JL ≅ KM.
- Use the segment addition postulate to show that JK + KL = JL and KL + LM = KM.
- Subtract KL from both sides of JK + KL = KL + LM to show that JK = LM.
- The first step is to show that JL ≅ KM. This is given in the problem.
The second step is to use the segment addition postulate to show that JK + KL = JL and KL + LM = KM. The segment addition postulate states that the sum of the lengths of two segments that have a common endpoint is equal to the length of the third segment that has the same endpoints.
In this case, JK + KL is the sum of the lengths of the segments that have endpoint J, and JL is the third segment that has endpoint J. Similarly, KL + LM is the sum of the lengths of the segments that have endpoint K, and KM is the third segment that has endpoint K.
The third step is to subtract KL from both sides of JK + KL = KL + LM to show that JK = LM. This is done using the subtraction property of equality, which states that if a = b, then a - c = b - c. In this case, we have JK + KL = KL + LM, so JK + KL - KL = KL + LM - KL. Simplifying both sides of the equation gives JK = LM.
The fourth step is to use the definition of congruence to conclude that JK ≅ LM. The definition of congruence states that two segments are congruent if and only if they have the same length. In this case, we have shown that JK = LM, so we can conclude that JK ≅ LM.
Therefore, we have proven that JK ≅ LM by showing that JL ≅ KM, using the segment addition postulate, and using the definition of congruence.
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