Question

use the given information to complete the proof of the following theorem.
if a quadrilateral is a parallelogram, then its opposite angles are congruent.
by definition, a parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
use this definition in your proof.
given: ( pqrs ) is a parallelogram
prove: ( angle p cong angle r )

statement	reason
2 ( overline{pq} parallel )	definition of a parallelogram
3 ( overline{ps} parallel )	definition of a parallelogram
4 ( angle p ) and ( angle ) are supp. angles	if lines ( parallel ), then cons. int. ( angle s ) are supp.
5 ( angle p ) and ( angle ) are supp. angles	if lines ( parallel ), then cons. int. ( angle s ) are supp.
6 ( angle q cong angle )	reason?
7 ( angle r ) and ( angle ) are supp. angles	reason?
8 ( angle p cong angle )	reason?

Explanation:

Step1: Fill in the blanks for parallel sides

Since \(PQRS\) is a parallelogram, by the definition of a parallelogram (\(PQRS\) has both pairs of opposite sides parallel), for statement 2, \(\overline{PQ}\parallel\overline{SR}\) (opposite sides of a parallelogram are parallel). For statement 3, \(\overline{PS}\parallel\overline{QR}\) (opposite sides of a parallelogram are parallel).

Step2: Identify supplementary angles

For statement 4, since \(\overline{PQ}\parallel\overline{SR}\) and \(\overline{PS}\) is a transversal, \(\angle P\) and \(\angle S\) are supplementary angles (If lines \(\parallel\), then consecutive - interior \(\angle s\) are supp.). For statement 5, since \(\overline{PS}\parallel\overline{QR}\) and \(\overline{PQ}\) is a transversal, \(\angle P\) and \(\angle Q\) are supplementary angles (If lines \(\parallel\), then consecutive - interior \(\angle s\) are supp.).

Step3: Use the congruent - angle property

For statement 6, \(\angle Q\cong\angle S\). The reason is: Angles supplementary to the same angle are congruent. Let \(\angle P + \angle Q=180^{\circ}\) and \(\angle P+\angle S = 180^{\circ}\), then \(\angle Q=\angle S\) (by the subtraction property of equality, \(\angle Q=180^{\circ}-\angle P\) and \(\angle S = 180^{\circ}-\angle P\)).
For statement 7, since \(\overline{PQ}\parallel\overline{SR}\) and \(\overline{QR}\) is a transversal, \(\angle R\) and \(\angle Q\) are supplementary angles (If lines \(\parallel\), then consecutive - interior \(\angle s\) are supp.).

Step4: Prove \(\angle P\cong\angle R\)

For statement 8, \(\angle P\cong\angle R\). The reason is: Angles supplementary to congruent angles are congruent. We know \(\angle P+\angle Q = 180^{\circ}\) and \(\angle R+\angle Q=180^{\circ}\), so \(\angle P=\angle R\) (by the subtraction property of equality, \(\angle P=180^{\circ}-\angle Q\) and \(\angle R=180^{\circ}-\angle Q\)).

Answer:

Statement	Reason
2 \(\overline{PQ}\parallel\overline{SR}\)	Definition of a Parallelogram
3 \(\overline{PS}\parallel\overline{QR}\)	Definition of a Parallelogram
4 \(\angle P\) and \(\angle S\) are supp. angles	If lines \(\parallel\), then cons. int. \(\angle s\) are supp.
5 \(\angle P\) and \(\angle Q\) are supp. angles	If lines \(\parallel\), then cons. int. \(\angle s\) are supp.
6 \(\angle Q\cong\angle S\)	Angles supplementary to the same angle are congruent
7 \(\angle R\) and \(\angle Q\) are supp. angles	If lines \(\parallel\), then cons. int. \(\angle s\) are supp.
8 \(\angle P\cong\angle R\)	Angles supplementary to congruent angles are congruent