Question

1. which triangles are congruent by asa? options: δade and δcbf, δcde and δabf, δadc and δcba, none of these are possible to pr

Explanation:

To determine which triangles are congruent by ASA (Angle - Side - Angle) criterion, we analyze each option:

1. For \(\triangle ADE\) and \(\triangle CBF\): We need to check if two angles and the included side are equal. From the diagram, there is no clear indication that the angles and included side required for ASA are equal between these two triangles.

1. For \(\triangle CDE\) and \(\triangle ABF\): Similarly, there is no obvious pair of angles and an included side that are equal to satisfy the ASA congruence criterion.

1. For \(\triangle ADC\) and \(\triangle CBA\):

• We know that \(AB = DC\) (from the markings on the sides \(AB\) and \(DC\) in the diagram, indicating they are equal in length).
• \(\angle DAC=\angle BCA\) (marked equal in the diagram).
• \(\angle DCA = \angle BAC\) (alternate - interior angles, since \(AB\parallel DC\) as \(AB = DC\) and the sides are parallel, and \(AC\) is a transversal). Also, the side \(AC\) is common to both \(\triangle ADC\) and \(\triangle CBA\). So, in \(\triangle ADC\) and \(\triangle CBA\), we have \(\angle DAC=\angle BCA\), \(AC = AC\) (common side, included between the two angles), and \(\angle DCA=\angle BAC\). Thus, by ASA congruence criterion, \(\triangle ADC\cong\triangle CBA\).

Answer:

\(\boldsymbol{\triangle ADC}\) and \(\boldsymbol{\triangle CBA}\) (the option corresponding to \(\triangle ADC\) and \(\triangle CBA\))