Question

1. error analysis zhang says △abc is congruent to △adc. explain the error in zhang’s work.

\\( \overline { a d } \cong \overline { c d } \\)
\\( \overline { a c } \cong \overline { a c } \\)
\\( \angle d c a \cong \angle b c a \\)
therefore,
\\( \triangle a b c \cong \triangle a d c \\) by sas

1. construct arguments given \\( \triangle a b c \cong \triangle x y z \\), use a rigid motion to prove theorem 4 - 4, corresponding parts of congruent triangles

Explanation:

Zhang's error is in the first step. In the Side - Angle - Side (SAS) congruence criterion, the angle must be the included angle between the two sides. Here, the sides \(AD\) and \(CD\) are not the sides that form the angle \(\angle DCA\) in \(\triangle ADC\) and the sides \(AB\) and \(BC\) (if we were to use the correct correspondence) are not considered. Also, \(\overline{AD}\cong\overline{CD}\) is not a valid side - side congruence for the triangles \(\triangle ABC\) and \(\triangle ADC\) as per the given figure. The correct sides for the SAS criterion should be \(AC\) (common side), \(\angle BCA\) and \(\angle DCA\) (given congruent angles) and \(BC\) and \(DC\) (but \(BC\) and \(DC\) are not given as congruent).

Answer:

Zhang incorrectly identified the sides for the SAS (Side - Angle - Side) congruence criterion. The sides \(AD\) and \(CD\) are not the sides adjacent to the given congruent angles \(\angle DCA\) and \(\angle BCA\) in the correct triangles for the SAS comparison.