Question

11/14 - triangle congruence schoology assignment
possible points: 1.15
prove the following statement using the given pieces of information.
given: \\( \overline { a b } \parallel \overline { d c } , \overline { a d } \parallel \overline { b c } \\)
prove: \\( \triangle a b c \cong \triangle c d a \\)

Explanation:

Step1: Alternate interior angles

Since \( \overline{AB}\parallel\overline{DC}\), by the Alternate Interior Angles Theorem, \( \angle BAC\cong\angle DCA\).

Step2: Alternate interior angles

Since \( \overline{AD}\parallel\overline{BC}\), by the Alternate Interior Angles Theorem, \( \angle ACB\cong\angle CAD\).

Step3: Reflexive property

For \( \overline{AC}\cong\overline{AC}\), it is by the Reflexive Property of Congruence.

Step4: ASA congruence

In \( \triangle ABC\) and \( \triangle CDA\), we have \( \angle BAC\cong\angle DCA\), \( \overline{AC}\cong\overline{AC}\), \( \angle ACB\cong\angle CAD\). By the Angle - Side - Angle (ASA) Congruence Theorem, \( \triangle ABC\cong\triangle CDA\).

Answer:

1. Alternate Interior Angles Theorem; 4. Alternate Interior Angles Theorem; 5. Reflexive Property of Congruence; 6. ASA (Angle - Side - Angle) Congruence Theorem.