Proving a Quadrilateral is a Parallelogram - Editable Geometry ... - Free Printable

It looks like you've uploaded an image containing several geometry theorems related to parallelograms. The image includes:

- A definition of a parallelogram.
- Four theorems about properties that prove a quadrilateral is a parallelogram.
- Each theorem has a diagram and blank spaces to fill in conclusions.

Let’s go through each section one by one, solve the problem, and explain the reasoning.

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🔷 1. Definition of a Parallelogram

> If a quadrilateral has both pairs of opposite sides parallel, then it is a parallelogram.

Diagram:
A quadrilateral ABCD with AB ∥ CD and AD ∥ BC.

Given:
AB ∥ CD and AD ∥ BC

Then:
ABCD is a parallelogram ✔

✔ This is the definition — if both pairs of opposite sides are parallel, it's a parallelogram.

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🔷 2. Theorem: Both Pairs of Opposite Sides Congruent

> If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Diagram:
Quadrilateral ABCD with AB ≅ CD and AD ≅ BC.

Given:
AB ≅ CD and AD ≅ BC

Then:
ABCD is a parallelogram ✔

✔ This is a standard congruence-based theorem: If both pairs of opposite sides are equal in length, the figure is a parallelogram.

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🔷 3. Theorem: One Pair of Opposite Sides Both Parallel and Congruent

> If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.

Diagram:
AB ∥ CD and AB ≅ CD (indicated by tick marks and arrows)

Given:
AB ∥ CD and AB ≅ CD

Then:
ABCD is a parallelogram ✔

✔ This is a powerful shortcut: if one pair of opposite sides is both parallel and congruent, then the other pair must also be parallel and congruent → it's a parallelogram.

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🔷 4. Theorem: One Pair of Opposite Angles Congruent

> If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Wait! The text says "If both pairs of opposite angles", but the diagram shows only one angle marked.

But let’s look at the diagram:
- ∠A = 115°
- ∠C = 115°
- So ∠A ≅ ∠C
- Also, ∠B = 65°, ∠D = 65° (implied by sum of angles = 360°)
- So both pairs of opposite angles are congruent.

Given:
∠A ≅ ∠C and ∠B ≅ ∠D

Then:
ABCD is a parallelogram ✔

✔ Because in any quadrilateral, if both pairs of opposite angles are equal, it must be a parallelogram.

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🔷 5. Theorem: Diagonals Bisect Each Other

> If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Diagram:
Diagonals AC and BD intersect at point E, with AE ≅ EC and BE ≅ ED (shown with tick marks).

Given:
Diagonals bisect each other (AE = EC and BE = ED)

Then:
ABCD is a parallelogram ✔

✔ This is a key property: if the diagonals cut each other into two equal parts, the quadrilateral is a parallelogram.

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🔷 6. Theorem: One Pair of Opposite Sides Both Congruent and Parallel

> If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.

This is identical to Theorem #3.

Diagram:
AB ∥ CD and AB ≅ CD

Given:
AB ∥ CD and AB ≅ CD

Then:
ABCD is a parallelogram ✔

✔ Same as Theorem 3 — this is a common way to prove a parallelogram.

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✔ Final Answers (Filling in the blanks):

| Theorem | Given | Then |
|--------|-------|------|
| 1 | AB ∥ CD, AD ∥ BC | ABCD is a parallelogram |
| 2 | AB ≅ CD, AD ≅ BC | ABCD is a parallelogram |
| 3 | AB ∥ CD, AB ≅ CD | ABCD is a parallelogram |
| 4 | ∠A ≅ ∠C, ∠B ≅ ∠D | ABCD is a parallelogram |
| 5 | Diagonals bisect each other | ABCD is a parallelogram |
| 6 | AB ∥ CD, AB ≅ CD | ABCD is a parallelogram |

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🧠 Summary of Key Theorems That Prove a Quadrilateral is a Parallelogram:

1. Both pairs of opposite sides are parallel → Parallelogram (Definition)
2. Both pairs of opposite sides are congruent → Parallelogram
3. One pair of opposite sides is both parallel and congruent → Parallelogram
4. Both pairs of opposite angles are congruent → Parallelogram
5. Diagonals bisect each other → Parallelogram

These are all valid criteria to prove a quadrilateral is a parallelogram.

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