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 )
1 ( pqrs ) is a parallelogram given
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 ) if cons. int. ( angle s ) are supp., then lines ( parallel )
7 ( angle r ) and ( angle q ) are supp. angles if cons. int. ( angle s ) are supp., then lines ( parallel )
8 ( angle p cong angle r ) reason?

Explanation:

Step1: Analyze the given information

We know that \(PQRS\) is a parallelogram (\(\overline{PQ}\parallel\overline{SR}\) and \(\overline{PS}\parallel\overline{QR}\)), \(\angle P+\angle S = 180^{\circ}\) (since \(\overline{PQ}\parallel\overline{SR}\), consecutive - interior angles are supplementary), \(\angle P+\angle Q=180^{\circ}\) (since \(\overline{PS}\parallel\overline{QR}\), consecutive - interior angles are supplementary), \(\angle Q\cong\angle S\) (from the previous two equations: if \(a + b=180^{\circ}\) and \(a + c = 180^{\circ}\), then \(b = c\)), \(\angle R+\angle Q = 180^{\circ}\) (since \(\overline{PQ}\parallel\overline{SR}\), consecutive - interior angles are supplementary).

Step2: Use the transitive property of congruence

We have \(\angle P+\angle Q=180^{\circ}\) and \(\angle R+\angle Q = 180^{\circ}\).
Let \(x=\angle P\), \(y = \angle Q\), \(z=\angle R\). Then \(x + y=180^{\circ}\) and \(z + y=180^{\circ}\).
By the subtraction property of equality, \(x=180^{\circ}-y\) and \(z=180^{\circ}-y\). So \(x = z\).
In terms of angles, if two angles (\(\angle P\) and \(\angle R\)) are supplementary to the same angle (\(\angle Q\)), then \(\angle P\cong\angle R\) (using the Congruent Supplements Theorem: If two angles are supplementary to the same angle, then the two angles are congruent).

Answer:

Congruent Supplements Theorem.