Conjecture: A quadrilateral with one pair of sides both congruent and parallel is a parallelogram.

Question

1. Draw a diagram of the situation. 2. Mark the given information. 3. Restate the conjecture as a specific statement using the diagram.

Answer 1

A

_____B

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---------C

D

Figure is not drawn to scale.

Given:

Line segment AB is congruent to line segment CD in quadrilateral ABCD, and line segment AB is parallel to line segment CD.

Proof:

Construct a diagonal from Point B to point D.

Since line segment AB is parallel to line segment CD, and the following line segments are transversals of those line segments: AD, BC, and BD; therefore ∠ABD ≅ ∠CDB and ∠ADB ≅∠CBD by the alternate interior angles theorem.

Line segment BC is congruent to itself by the reflexive property of the congruence operator.

By the ASA postulate, triangles ABD and CDB are congruent. So line segment AD is congruent to line segment BC, because they are corresponding parts of congruent triangles.

Since both pairs of opposite sides are congruent, quadrilateral ABCD is a parallelogram.

(The congruent angles referenced also prove that line segments BC and AD are parallel by the converse of the alternate interior angles theorem.)

Answer 2

Hi Keiry,

Conjecture: A quadrilateral with one pair of sides both congruent and parallel is a parallelogram.

Diagram Description: Imagine a quadrilateral ABCD where AB is parallel to CD and AB is congruent to CD.

Markings:

1. Arrows on AB and CD to indicate they are parallel.
2. Tick marks on AB and CD to show they are congruent.

Restated Conjecture: Given a quadrilateral ABCD where AB is parallel to CD and AB is congruent to CD, then ABCD is a parallelogram.

Proof

Step 1: Definitions

1. In a parallelogram, opposite sides are parallel and opposite angles are congruent.

Step 2: Given

1. AB is parallel to CD (Given)
2. AB is congruent to CD (Given)

Step 3: Prove Opposite Sides Are Parallel

1. Draw diagonal AC in quadrilateral ABCD.
2. By the Alternate Interior Angles Theorem, angle BAC is congruent to angle ACD since AB is parallel to CD.
3. Triangle ABC and triangle ADC share angle CAD.
4. Triangle ABC and triangle ADC also share side AC.
5. By the ASA (Angle-Side-Angle) Congruence Theorem, triangle ABC is congruent to triangle ADC.
6. Angles ACD and BCA are corresponding angles between these congruent triangles, so they are congruent.
7. Since angle ACD is congruent to angle BCA and AB is parallel to CD, then BC must be parallel to AD by the Converse of the Alternate Interior Angles Theorem.
8. Now we have AB parallel to CD and BC parallel to AD, making ABCD a parallelogram.

Step 4: Conclusion

1. ABCD is a parallelogram because its opposite sides are parallel.

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Best regards,

Ben