https://file.aiquickdraw.com/mathbot/file/1729838417602xvq4jz4t.jpg

This proof involves showing that ∠P is congruent to ∠R using properties of parallelograms. Let's break it down step-by-step and fill in the missing reasons.

### Given:
- \( PQRS \) is a parallelogram.

### To Prove:
- \( \angle P \cong \angle R \).

### Proof:

| Statement | Reason |
| --------- | ------- |
| 1. \( PQRS \) is a parallelogram | Given |
| 2. \( PQ \parallel SR \) | Definition of a Parallelogram |
| 3. \( PS \parallel QR \) | Definition of a Parallelogram |
| 4. \( \angle P \) and \( \angle S \) are supplementary angles | If lines are parallel, then consecutive interior angles are supplementary. |
| 5. \( \angle P \) and \( \angle R \) are supplementary angles | If lines are parallel, then consecutive interior angles are supplementary. |
| 6. \( \angle Q \cong \angle S \) | Opposite angles in a parallelogram are congruent. |
| 7. \( \angle R \) and \( \angle S \) are supplementary angles | If two angles are supplementary to the same angle, they are congruent. |
| 8. \( \angle P \cong \angle R \) | If two angles are supplementary to the same angle, they are congruent. |

### Explanation:
1. The first three steps establish that we are dealing with a parallelogram and state its properties (both pairs of opposite sides are parallel).
2. Step 4 and Step 5 apply the consecutive interior angles theorem (since the sides are parallel, the consecutive interior angles \( \angle P \) and \( \angle S \), and \( \angle P \) and \( \angle R \), are supplementary).
3. In Step 6, opposite angles in a parallelogram are congruent, meaning \( \angle Q \cong \angle S \).
4. Finally, since \( \angle P \) and \( \angle R \) are both supplementary to \( \angle S \), we conclude that \( \angle P \cong \angle R \) by the supplementary angle property (if two angles are supplementary to the same angle, they are congruent).

Would you like more details about any specific part of the proof?

### 5 Related Questions:
1. What is the significance of the consecutive interior angle theorem in parallelograms?
2. Why are opposite angles in a parallelogram congruent?
3. How do supplementary angles relate to congruent angles in this proof?
4. Can this proof be applied to all quadrilaterals, or just parallelograms?
5. What other properties of parallelograms can be useful in geometric proofs?

### Tip:
In parallelogram proofs, always keep in mind the key properties: opposite sides are parallel and equal, opposite angles are congruent, and consecutive angles are supplementary.

Prove that angle P is congruent to angle R in parallelogram PQRS.

Explore the proof of congruence between angle P and angle R in parallelogram PQRS using the properties of parallelograms, including the consecutive interior angle theorem and opposite angles theorem. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation