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Teacher Notes

CONCEPT 1 - Prove theorems about parallelograms.

 TEACHER NOTE -- I have provided a number of different ways to prove these concepts, both traditional and transformational. This allows for flexibility on your part to determine the rigor and style of the proof.

(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.

 TEACHER NOTE -- Notice how the symmetry that we established in G.CO.3 is showing up here to help us establish properties of parallelograms.

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.

 TEACHER NOTE -- The converse arguement on these is essential. Identifying when something is a parallelogram is a major help to proving many other concepts. We will also use these converse arguements when doing coordinate proof to determine the type of quadrilateral that we are looking at.

(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.