Question

given:
parallelogram abcd
prove:
opposite sides are congruent.
(ad≅bc and ab≅dc)
(draw an auxiliary segment bd)

Explanation:

Step1: Properties of parallelogram

In parallelogram \(ABCD\), \(AB\parallel DC\) and \(AD\parallel BC\) (Definition of parallelogram).

Step2: Alternate - interior angles

\(\angle ABD\cong\angle CDB\) (Alternate - interior angles, since \(AB\parallel DC\) and \(BD\) is a transversal). \(\angle ADB\cong\angle CBD\) (Alternate - interior angles, since \(AD\parallel BC\) and \(BD\) is a transversal).

Step3: Common side

\(BD = BD\) (Reflexive property of congruence).

Step4: ASA congruence

\(\triangle ABD\cong\triangle CDB\) (By ASA (Angle - Side - Angle) congruence criterion: \(\angle ABD\cong\angle CDB\), \(BD = BD\), \(\angle ADB\cong\angle CBD\)).

Step5: Corresponding parts of congruent triangles

\(AD\cong BC\) and \(AB\cong DC\) (Corresponding parts of congruent triangles (\(\triangle ABD\) and \(\triangle CDB\)) are congruent).

Answer:

Statements	Reasons
\(\angle ABD\cong\angle CDB\), \(\angle ADB\cong\angle CBD\)	Alternate - interior angles
\(BD = BD\)	Reflexive property
\(\triangle ABD\cong\triangle CDB\)	ASA congruence
\(AD\cong BC\), \(AB\cong DC\)	Corresponding parts of congruent triangles