Question

try again
your answer is incorrect.

• statement 6: your answer is incorrect.
• statement 7: your answer is incorrect.
• statement 8: your answer is incorrect.

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: ∠p ≅ ∠r
1 pqrs is a parallelogram given
2 \\( \overline { p q } \parallel \overline { s r } \\) definition of a parallelogram
3 \\( \overline { p s } \parallel \overline { q r } \\) definition of a parallelogram
4 ∠p and ∠s are supp. angles if lines ∥, then cons. int. ∠s are supp.
5 ∠p and ∠q are supp. angles if lines ∥, then cons. int. ∠s are supp.
6 ∠q ≅ ∠s if cons. int. ∠s are supp., then lines ∥
7 ∠r and ∠q are supp. angles if cons. int. ∠s are supp., then lines ∥
8 ∠p ≅ ∠r definition of congruent angles

Explanation:

Step1: Analyze statement 6

If two angles are supplementary to the same angle, then they are congruent. Since \(\angle P+\angle S = 180^{\circ}\) (statement 4) and \(\angle P+\angle Q=180^{\circ}\) (statement 5), by the congruent supplements theorem, \(\angle Q\cong\angle S\).

Step2: Analyze statement 7

Since \(PS\parallel QR\) (statement 3), by the property that if two parallel lines are cut by a transversal, then consecutive - interior angles are supplementary. So \(\angle R\) and \(\angle Q\) are supplementary angles.

Step3: Analyze statement 8

We know that \(\angle P+\angle Q = 180^{\circ}\) (statement 5) and \(\angle R+\angle Q=180^{\circ}\) (statement 7). By the congruent supplements theorem (if two angles are supplementary to the same angle, then they are congruent), \(\angle P\cong\angle R\).

Answer:

• Statement 6: \(\angle Q\cong\angle S\), Reason: Congruent Supplements Theorem
• Statement 7: \(\angle R\) and \(\angle Q\) are supp. angles, Reason: If lines \(\parallel\), then cons. int. \(\angle\)s are supp.
• Statement 8: \(\angle P\cong\angle R\), Reason: Congruent Supplements Theorem