Quadrilaterals Worksheets - Math Monks - Free Printable

Let's go through each problem on the "Theorems Involving Quadrilaterals Worksheet" and solve them step by step using geometric properties of quadrilaterals, especially parallelograms and rectangles.

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Problem 1:

Given: PQRS is a parallelogram
Prove:
(a) $ \overline{PQ} \cong \overline{RS} $
(b) $ \overline{PS} \cong \overline{RQ} $

#### ✔ Solution:

In a parallelogram, opposite sides are congruent.

- Since PQRS is a parallelogram:
- Opposite sides are equal in length.
- Therefore:
- (a) $ \overline{PQ} \cong \overline{RS} $
- (b) $ \overline{PS} \cong \overline{RQ} $

✔ Conclusion: By the Opposite Sides Theorem for parallelograms, both statements are true.

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Problem 2:

Given: JKLM is a parallelogram
Prove:
$ \angle J \cong \angle L $, $ \angle M \cong \angle K $

#### ✔ Solution:

In a parallelogram, opposite angles are congruent.

- Since JKLM is a parallelogram:
- $ \angle J $ and $ \angle L $ are opposite angles → $ \angle J \cong \angle L $
- $ \angle M $ and $ \angle K $ are opposite angles → $ \angle M \cong \angle K $

✔ Conclusion: By the Opposite Angles Theorem, these pairs are congruent.

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Problem 3:

Given: EFGH is a parallelogram
Prove: $ \angle E $ and $ \angle F $ are supplementary

#### ✔ Solution:

In a parallelogram, consecutive (adjacent) angles are supplementary (add up to 180°).

- $ \angle E $ and $ \angle F $ are adjacent angles.
- So: $ \angle E + \angle F = 180^\circ $
- Therefore, they are supplementary

✔ Conclusion: By the Consecutive Angles Theorem, adjacent angles in a parallelogram are supplementary.

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Problem 4:

Given: ABCD and CDRS are parallelograms
Prove: $ \overline{AB} \cong \overline{SR} $

#### ✔ Solution:

We are given two parallelograms:
- ABCD is a parallelogram → $ \overline{AB} \cong \overline{CD} $ (opposite sides)
- CDRS is a parallelogram → $ \overline{CD} \cong \overline{SR} $ (opposite sides)

Now use transitive property of congruence:

- $ \overline{AB} \cong \overline{CD} $
- $ \overline{CD} \cong \overline{SR} $
- So: $ \overline{AB} \cong \overline{SR} $

✔ Conclusion: $ \overline{AB} \cong \overline{SR} $ by transitivity.

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Problem 5:

Given: UVWX is a rectangle
Prove:
(a) $ \overline{UV} \parallel \overline{WX} $
(b) $ \overline{UW} \parallel \overline{VX} $

#### ✔ Solution:

A rectangle is a special type of parallelogram with four right angles.

So, since UVWX is a rectangle:
- It is a parallelogram → opposite sides are parallel
- Also, all angles are 90°

Thus:
- (a) $ \overline{UV} \parallel \overline{WX} $ — opposite sides of a parallelogram are parallel
- (b) $ \overline{UW} \parallel \overline{VX} $ — same reason, opposite sides are parallel

✔ Conclusion: Both pairs of opposite sides are parallel because a rectangle is a parallelogram.

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Problem 6:

Given: ABCD is a parallelogram
Prove:
(a) $ \overline{AB} \cong \overline{DC} $
(b) $ \overline{AD} \cong \overline{BC} $

#### ✔ Solution:

This is again based on the properties of parallelograms.

In any parallelogram:
- Opposite sides are congruent.

So:
- (a) $ \overline{AB} \cong \overline{DC} $ → opposite sides
- (b) $ \overline{AD} \cong \overline{BC} $ → opposite sides

✔ Conclusion: These follow directly from the Opposite Sides Congruence Theorem.

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🔚 Final Summary:

| Problem | Key Theorem Used | Conclusion |
|--------|------------------|-----------|
| 1 | Opposite sides of a parallelogram are congruent | $ PQ \cong RS $, $ PS \cong RQ $ |
| 2 | Opposite angles of a parallelogram are congruent | $ \angle J \cong \angle L $, $ \angle M \cong \angle K $ |
| 3 | Consecutive angles in a parallelogram are supplementary | $ \angle E + \angle F = 180^\circ $ |
| 4 | Transitive property of congruence + opposite sides of parallelograms | $ AB \cong SR $ |
| 5 | Rectangles are parallelograms → opposite sides parallel | $ UV \parallel WX $, $ UW \parallel VX $ |
| 6 | Opposite sides of a parallelogram are congruent | $ AB \cong DC $, $ AD \cong BC $ |

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✔ All problems solved using fundamental quadrilateral theorems, particularly those involving parallelograms and rectangles.

Let me know if you'd like formal two-column proofs for any of these!