Question

Determine whether this conjecture is true or false. Give a counterexample for any false conjectures. If DE ⊥ EF, then DEF is a right angle

Answer 1

The conjecture 'If DE ⊥ EF, then DEF is a right angle' is false, as demonstrated by considering a three-dimensional scenario where angle DEF could be acute or obtuse, even when DE is perpendicular to EF.

Step-by-step explanation:

When examining the conjecture "If DE ⊥ EF, then DEF is a right angle," we are analyzing a statement in geometry regarding the properties of angles and lines. To determine if this conjecture is true or false, we can employ the concept of a counterexample. A counterexample in mathematics is an example that disproves a conjecture by showing that even though the premises are true, the conclusion is false.

To demonstrate a counterexample for this conjecture, consider the following scenario: Suppose DE and EF are indeed perpendicular lines (lines that intersect to form a 90-degree angle). For the conjecture to hold true, it would mean that any angle formed at the intersection of DE and EF must be a right angle. However, this is not necessarily the case. If we look at a three-dimensional scenario, where DE is perpendicular to EF, but not in the plane that contains angle DEF, it is entirely possible for angle DEF to be acute or obtuse and not a right angle. Therefore, this serves as a counterexample, showing that the original conjecture is false.

Answer 2

If DE is perpendicular to EF, it does not necessarily mean that DEF is a right angle.

Step-by-step explanation:

In this case, we are given that DE is perpendicular to EF. To determine whether DEF is a right angle, we need to examine the relationship between DE and EF. If DE and EF are not perpendicular, then DEF will not be a right angle.

A counterexample to this conjecture would be if DE and EF are not perpendicular, but DEF is still a right angle. For example, if DE and EF are parallel to each other, then DEF would be a right angle even though DE is not perpendicular to EF.

Therefore, the conjecture that if DE is perpendicular to EF, then DEF is a right angle is false.