Question

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

\\(\overline{xw} \cong \overline{zw}\\) | congruent sides on a straight line
\\(\angle xyw \cong \angle zyw\\) | perpendicular angles on a straight line
\\(\overline{yw} \cong \overline{yw}\\) | common side
\\(\therefore \delta xyw \cong \delta zyw\\) | asa

Explanation:

Analyze the first statement and reason

Using the Triangle Congruence knowledge point

• Statement: \(\overline{XW} \cong \overline{ZW}\)
• Reason: "Congruent sides on a straight line"
• Analysis: The diagram shows tick marks on \(\overline{XY}\) and \(\overline{ZY}\), meaning \(\overline{XY} \cong \overline{ZY}\). There is no information given that \(\overline{XW} \cong \overline{ZW}\). Furthermore, the reason "Congruent sides on a straight line" is mathematically invalid. Thus, this step is a mistake.

Analyze the second statement and reason

Using the Triangle Congruence knowledge point

• Statement: \(\angle XYW \cong \angle ZYW\)
• Reason: "Perpendicular angles on a straight line"
• Analysis: The diagram shows \(\overline{YW} \perp \overline{XZ}\) at point \(Y\). Since perpendicular lines form right angles, \(\angle XYW\) and \(\angle ZYW\) are both right angles, making them congruent. While the phrasing of the reason is slightly informal, the geometric deduction itself is valid.

Analyze the third statement and reason

Using the Triangle Congruence knowledge point

• Statement: \(\overline{YW} \cong \overline{YW}\)
• Reason: "Common Side"
• Analysis: This is a correct application of the Reflexive Property of Congruence for the shared side \(\overline{YW}\).

Analyze the final congruence statement and reason

Using the Triangle Congruence knowledge point

• Statement: \(\triangle XYW \cong \triangle ZYW\)
• Reason: "ASA"
• Analysis: Based on the valid parts of the diagram, we have:

1. \(\overline{XY} \cong \overline{ZY}\) (Side)
2. \(\angle XYW \cong \angle ZYW\) (Angle)
3. \(\overline{YW} \cong \overline{YW}\) (Side)

• This corresponds to the SAS (Side-Angle-Side) congruence criterion, not ASA (Angle-Side-Angle). Therefore, using "ASA" as the reason is a mistake.

Answer:

The mistakes in the two-column proof are:

1. First Row Mistake:

• Statement: \(\overline{XW} \cong \overline{ZW}\) is incorrect because the given congruent segments in the diagram are \(\overline{XY} \cong \overline{ZY}\).
• Reason: "Congruent sides on a straight line" is not a valid geometric postulate or theorem.

1. Fourth Row Mistake:

• Reason: "ASA" is incorrect. The correct congruence criterion based on the established congruent parts—\(\overline{XY} \cong \overline{ZY}\) (Side), \(\angle XYW \cong \angle ZYW\) (Angle), and \(\overline{YW} \cong \overline{YW}\) (Side)—is SAS (Side-Angle-Side).