Question

given:
parallelogram abcd
prove:
opposite angles are congruent. (\\( \angle a \cong \angle c \\))
(draw an auxiliary segment \\( b d \\))

Explanation:

Step1: Definition of parallelogram

In parallelogram \(ABCD\), \(AB\parallel CD\) and \(AB = CD\), \(AD\parallel BC\) and \(AD=BC\) (by the definition of a parallelogram).

Step2: Alternate - interior angles

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

Step3: ASA congruence

In \(\triangle ABD\) and \(\triangle CDB\), we have \(AB = CD\) (from parallelogram properties), \(\angle ABD=\angle CDB\) (from step 2), \(\angle ADB=\angle CBD\) (from step 2). So, \(\triangle ABD\cong\triangle CDB\) (by ASA - Angle - Side - Angle congruence criterion).

Step4: Corresponding parts of congruent triangles

Since \(\triangle ABD\cong\triangle CDB\), \(\angle A\cong\angle C\) (corresponding parts of congruent triangles are congruent).

Answer:

Statements	Reasons
\(\angle ABD\cong\angle CDB\), \(\angle ADB\cong\angle CBD\)	Alternate - interior angles (for parallel lines \(AB\parallel CD\) and \(AD\parallel BC\) with transversal \(BD\))
\(\triangle ABD\cong\triangle CDB\)	ASA congruence criterion
\(\angle A\cong\angle C\)	Corresponding parts of congruent triangles (\(\triangle ABD\) and \(\triangle CDB\))