Question

question 6
the following is an incomplete paragraph proving that the opposite sides of parallelogram abcd are congruent.
according to the given information, ab || dc and bc || ad. construct a diagonal from a to c with a straightedge. it is congruent to itself by the reflexive property of equality. angles bac and dca are congruent by the alternate interior angles theorem. angles bca and dac are congruent by the alternate interior theorem. ________ by cpctc, opposite sides ab and cd, as well as sides bc and da, are congruent.
which statement accurately completes the proof?
triangles bca and dac are congruent according to the angle - angle - side (aas) postulate.
triangles bca and dac are congruent according to the angle - side - angle (asa) postulate.
angles abc and cda are congruent according to a property of parallelograms (opposite angles are congruent).
angles bad and adc as well as angles dcb and cba are supplementary by the same - side interior angles theorem.

Explanation:

Step1: Recall congruence postulates

We have two - pairs of congruent angles (alternate - interior angles) and a common side (diagonal AC).

Step2: Identify the congruence postulate

The Angle - Side - Angle (ASA) postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. In \(\triangle BCA\) and \(\triangle DAC\), \(\angle BAC\cong\angle DCA\), \(AC = AC\) (reflexive property), and \(\angle BCA\cong\angle DAC\). So, \(\triangle BCA\cong\triangle DAC\) by ASA.

Answer:

Triangles BCA and DAC are congruent according to the Angle - Side - Angle (ASA) Postulate.