Question

1. error analysis zhang says \\( \triangle a b c \\) is congruent to \\( \triangle a d c \\). explain the error in zhangs 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 13. 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 are congruent. 14. mathematical connections is \\( \triangle j k l \\) congruent to \\( \triangle m n l \\) ? explain. 17. prove \\( \triangle a c e \\) is an isosceles triangle. see examples 1 and 2 18. what is \\( m \angle r t s \\) ? justify your answer. see examples 1 and 2 19. what additional information is needed to show that \\( \triangle p q r \cong \triangle s t u \\) by sss? see example 3

Explanation:

Step1: Recall the SAS (Side - Angle - Side) congruence criterion

The SAS congruence criterion states that for two triangles \(\triangle ABC\) and \(\triangle ADC\), if two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle, then the triangles are congruent.

Step2: Identify the sides and angles in Zhang's claim

In Zhang's work, \(\overline{AD}\cong\overline{CD}\), \(\overline{AC}\cong\overline{AC}\), and \(\angle DCA\cong\angle BCA\). But in the SAS criterion, the angle must be the included angle between the two sides. For \(\triangle ABC\) and \(\triangle ADC\), in the SAS formula \( (side - angle - side)\), for \(\triangle ABC\) the sides are \(\overline{BC}\) and \(\overline{AC}\) with included angle \(\angle BCA\), and for \(\triangle ADC\) the sides are \(\overline{AD}\) and \(\overline{AC}\) with included angle \(\angle DCA\). The sides \(\overline{AD}\) and \(\overline{BC}\) are not known to be congruent.

Answer:

Zhang's error is that in the SAS congruence criterion, the angle must be the included angle between the two pairs of congruent sides. Here, \(\overline{AD}\) and \(\overline{BC}\) are not shown to be congruent, so \(\triangle ABC
ot\cong\triangle ADC\) by SAS.