Question

1. fill in the blanks to make the proof that triangle abc is congruent to triangle cda. it is given that quadrilateral abcd is a parallelogram.

a. line ab is parallel to line dc and cut by transversal ac, so angles cab and acd are ____ and must be congruent.
b. line ad is parallel to line bc and cut by transversal ac, so angles acb and cad are ____ and must be congruent.

Explanation:

Step1: Recall parallel - line properties

When two parallel lines are cut by a transversal, alternate interior angles are congruent.
For line \(AB\parallel DC\) and transversal \(AC\), by the alternate - interior angles theorem, \(\angle CAB\) and \(\angle ACD\) are congruent.
For line \(AD\parallel BC\) and transversal \(AC\), by the alternate - interior angles theorem, \(\angle ACB\) and \(\angle CAD\) are congruent.
Also, in a parallelogram \(ABCD\), \(AB = DC\) and \(AD = BC\).

Step2: Prove \(\triangle ABC\cong\triangle CDA\)

In \(\triangle ABC\) and \(\triangle CDA\):

• \(\angle CAB=\angle ACD\) (from \(AB\parallel DC\) and transversal \(AC\))
• \(AC = CA\) (common side)
• \(\angle ACB=\angle CAD\) (from \(AD\parallel BC\) and transversal \(AC\))

By the ASA (Angle - Side - Angle) congruence criterion, \(\triangle ABC\cong\triangle CDA\).

Answer:

a. alternate - interior angles; b. alternate - interior angles