Question

the two-column proof to prove \\(\delta efd \cong \delta nmo\\) is given below, but there are errors. select each mistake that exists in the proof.

\\(\

$$\begin{array}{|c|c|} \\hline \\overline{ef} \\cong \\overline{nm} & \\text{given} \\\\ \\hline \\overline{df} \\cong \\overline{om} & \\text{hypotenuses are always congruent} \\\\ \\hline \\angle def \\cong \\angle onm & \\perp \\text{ angles are congruent} \\\\ \\hline \\therefore \\delta efd \\cong \\delta nmo & \\text{sas} \\\\ \\hline \\end{array}$$

\\)

Explanation:

Analyze the given proof steps

Using the Two-Column Proof Errors knowledge point:

1. Row 1: \(\overline{EF} \cong \overline{NM}\) with reason "Given".

• Looking at the diagram, both segments \(\overline{EF}\) and \(\overline{NM}\) have a single tick mark. Thus, they are given as congruent. This step is correct.

1. Row 2: \(\overline{DF} \cong \overline{OM}\) with reason "Hypotenuses are always congruent".

• Looking at the diagram, both hypotenuses \(\overline{DF}\) and \(\overline{OM}\) have double tick marks, meaning they are given as congruent.
• However, the reason "Hypotenuses are always congruent" is mathematically incorrect. Hypotenuses of two different right triangles are not always congruent; they are congruent here because it is given in the diagram. The correct reason should be "Given".

1. Row 3: \(\angle DEF \cong \angle ONM\) with reason "\(\perp\) angles are congruent".

• Looking at the diagram, \(\angle DEF\) and \(\angle ONM\) are both right angles (indicated by square symbols).
• All right angles are congruent. The reason "\(\perp\) angles are congruent" (or "All right angles are congruent") is correct.

1. Row 4: \(\therefore \Delta EFD \cong \Delta NMO\) with reason "SAS".

• Using the Triangle Congruence knowledge point:
• We have established congruence for:
• Side: \(\overline{EF} \cong \overline{NM}\)
• Side: \(\overline{DF} \cong \overline{OM}\) (Hypotenuse)
• Angle: \(\angle DEF \cong \angle ONM\) (Right angle)
• The angle is NOT the included angle between the two congruent sides. The included angle between \(\overline{EF}\) and \(\overline{DF}\) is \(\angle DFE\).
• Since the congruent angle is not included, "SAS" (Side-Angle-Side) is incorrect.
• Because these are right triangles, the correct congruence theorem to use with a congruent leg and hypotenuse is HL (Hypotenuse-Leg). Thus, "SAS" is a mistake.

Answer:

The mistakes in the proof are:

1. The reason in Row 2 is incorrect: The reason given is "Hypotenuses are always congruent", but the correct reason is Given (as indicated by the matching tick marks on the diagram).
2. The reason in Row 4 is incorrect: The reason given is "SAS", but the correct congruence theorem is HL (Hypotenuse-Leg) because the congruent angle is a right angle and is not the included angle between the congruent sides.