Question

1. by definition, a parallelogram is a quadrilateral in which both pairs of opposite sides

use this definition in your proof.
given: ( a b c d ) is a parallelogram.
prove: ( overline{a d} cong overline{b c} )

1. for each part below, use the figure to fill in the blank.

Explanation:

Step1: Identify parallel sides

Since \(ABCD\) is a parallelogram, by the definition of a parallelogram, \(\overline{AB}\parallel\overline{DC}\) and \(\overline{AD}\parallel\overline{BC}\).

Step2: Find congruent angles

When \(\overline{AB}\parallel\overline{DC}\) and \(AC\) is a transversal, \(\angle1\cong\angle3\) (alternate - interior angles). When \(\overline{AD}\parallel\overline{BC}\) and \(AC\) is a transversal, \(\angle2\cong\angle4\) (alternate - interior angles).

Step3: State the common side

\(\overline{AC}\cong\overline{AC}\) (reflexive property of congruence).

Step4: Prove triangle congruence

By the ASA (Angle - Side - Angle) congruence criterion, \(\triangle ABC\cong\triangle CDA\) (because \(\angle1\cong\angle3\), \(\overline{AC}\cong\overline{AC}\), \(\angle2\cong\angle4\)).

Step5: Conclude side congruence

Since \(\triangle ABC\cong\triangle CDA\), by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \(\overline{AD}\cong\overline{BC}\).

Answer:

1. \(\angle3\)
2. \(\angle4\)
3. \(\overline{AC}\cong\overline{AC}\); Reflexive Property of Congruence
4. \(CDA\); ASA (Angle - Side - Angle)
5. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)