CONCEPT 1 - Prove theorems about parallelograms.

(1) Prove that opposite sides of a parallelogram are congruent.

a) Proof by Symmetry and Patty Paper (Informal – Transformational Approach)

b) Proof by Congruent Triangles (Formal – Classic Approach)

(2) Prove that opposite angles of a parallelogram are congruent.

a) Proof by Symmetry and Patty Paper (Informal – Transformational Approach)

b) Proof by Congruent Triangles (Formal – Classic Approach)

(3) Prove that diagonals bisect each other in a parallelogram.

a) Proof by Symmetry and Patty Paper (Informal – Transformational Approach)

b) Proof by Congruent Triangles (Formal – Classic Approach)

4. Prove that diagonals are congruent in a rectangle.

a) Proof by Symmetry and Patty Paper (Informal – Transformational Approach)

b) Proof by Triangle Congruence (Formal – Classic Approach)

5. Prove that the diagonals of a rhombus are angle bisectors.

a) Proof by Symmetry and Patty Paper (Informal – Transformational Approach)

b) Proof by Triangle Congruence (Formal – Classic Approach)

6. Prove that the diagonals of a rhombus are perpendicular.

a) Proof by Symmetry and Patty Paper (Informal – Transformational Approach)

b) Proof by Triangle Congruence (Formal – Classic Approach)

CONCEPT 2 - Conversely, Establish when a quadrilateral is a parallelogram.

(1) If a quadrilateral has two sets of opposite parallel sides.

By definition, it must be a parallelogram.

(2) If a quadrilateral has two sets of opposite congruent sides.

 (3) If a quadrilateral has two sets of opposite congruent angles.

(4) If a quadrilateral has consecutive interior angles that are supplementary.

(5) If a quadrilateral has diagonals that bisect each other.

(6) if a quadrilateral has one set of opposite sides that are BOTH parallel and congruent.

TO ESTABLISH IF A QUADRILATERAL IS A PARALLELOGRAM

• Both pairs of opposite sides are parallel.
• Both pairs of opposite sides are congruent.
• Both pairs of opposite angles are congruent.
• Consecutive angles are supplementary.
• Diagonals bisect each other.
• One pair of opposite sides is both congruent and parallel.