Question

for each figure, use rigid motions to verify that the triangles are congruent. state the congruence theorem that you used.
a. given: parallelogram ( a b c d ) and ( overline{a b} parallel overline{c d} )
verify that ( \triangle a b c cong \triangle c d a ).

Explanation:

Step1: Use the property of parallelogram

In parallelogram \(ABCD\), \(AB = CD\) (opposite sides of a parallelogram are equal) and \(AD=BC\) (opposite sides of a parallelogram are equal). Also, since \(AB\parallel CD\), \(\angle BAC=\angle DCA\) (alternate - interior angles).

Step2: Apply the SAS congruence theorem

In \(\triangle ABC\) and \(\triangle CDA\), we have \(AB = CD\) (from parallelogram property), \(\angle BAC=\angle DCA\) (alternate - interior angles), and \(AC = CA\) (common side). By the Side - Angle - Side (SAS) congruence theorem, \(\triangle ABC\cong\triangle CDA\).

Answer:

\(\triangle ABC\cong\triangle CDA\) by the SAS (Side - Angle - Side) congruence theorem.