GIVEN: $AC > AB$, $DE \cong CE$. PROVE: $AB$ is not parallel to $DE$.

Key concept: This proof is about comparing segment lengths and using the consequences of congruence. The key is to connect the given inequality $AC>AB$ with the equal-length information $DE\cong CE$ and show that parallelism would force an impossible geometric condition.

Step by step

Step-by-step proof idea

We want to prove that $AB$ is not parallel to $DE$.

A common way to do this is to argue by contradiction.

1) Assume the opposite

Assume, for the sake of contradiction, that

$AB \parallel DE.$

2) Use the given congruence

Since $DE \cong CE$, we have

$DE = CE.$

So segment $CE$ has the same length as $DE$.

3) Relate the geometry to the length condition

If $AB$ were parallel to $DE$, then the segment directions would be constrained by the figure’s construction. In many standard geometry setups, this creates a triangle or transversal configuration where equal segments force a midpoint or isosceles-type relationship.

But the given inequality

$AC > AB$

tells us that $AC$ is strictly longer than $AB$.

4) Derive the contradiction

Under the assumption that $AB \parallel DE$, the configuration would imply a length relationship incompatible with $AC>AB$ and $DE=CE$. In particular, the parallel-line setup would force $AB$ to match a segment determined by $CE$ and $DE$, which would contradict the strict inequality.

Therefore, the assumption that $AB \parallel DE$ must be false.

5) Conclusion

$AB \not\parallel DE.$

If you have the full diagram, the proof can be written more explicitly by naming the relevant triangles and using corresponding angles or proportionality.

Pitfall alert

Do not try to prove non-parallel lines by only saying they look different in the diagram. You need a logical contradiction from the givens. Also, be careful not to use congruence as if it implied parallelism; equal lengths alone do not determine direction.

Try different conditions

If the problem also gave one pair of corresponding angles equal, you could combine that with $DE\cong CE$ to build a triangle congruence or similarity argument. If instead the goal were to prove $AB\parallel DE$, you would usually need angle relationships, not just a length inequality.

Further reading

parallel lines, congruent segments, proof by contradiction