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OBJECTIVE - G.CO.C.11

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

INTERPRETATION OF OBJECTIVE - G.CO.C.11

This is an objective needs very little interpretation. The purpose of this objective is to prove the properties of parallelograms. Remember that rectangles, rhombi and squares are types of parallelograms. These properties are essential for proving other things in the year. Notice trapezoids and kites are not mentioned; this might be because their properties are not used really in any other context, whereas using parallelogram properties is a very common strategy for proof and solving problems.

DIRECTLY IMPLIED SKILLS

(1) The student will be able to prove properties of parallelograms and then apply them.

(2) The student will be able to prove the properties of rectangles and then apply them.

(3) The student will be able to prove the properties of rhombi and then apply them.

(4) The student will be able to prove the properties of squares and then apply them.

(5) The student will be able to classify a quadrilateral by its properties.

INDIRECTLY IMPLIED SKILLS

(6) The student will be able to identify the conditions necessary to prove that a quadrilateral is a parallelogram.

THE BIG IDEA

A natural extension to triangle properties and relationships is the study of quadrilaterals.  This is because every quadrilateral can be divided into triangles. We will see how the earlier principles of symmetry and triangle congruence will unlock many of these relationships.

 Given Parallelogram ABCD   Due to rotational symmetry of 180 degrees (established in G.CO.2), we are able to map ∠A onto ∠C and ∠B onto ∠D, thus opposite angles of a parallelogram are congruent.

TRAPS & PITFALLS

The only pitfall I can think of is how difficult it is for the students to distinguish which properties are related to which quadrilaterals. I think it important for students to experience these properties from a number of different standpoints.

PAST CONNECTIONS

Students need to have a strong grasp on the definitions and properties of isometric transformations so they can establish relationships through transformations. They also need to know the newly established triangle congruence criteria so that they can be used in proof. Guess what they also need to know the relationships relating to parallel lines and a transversal. Notice that the early part of this unit has all been preparatory material so that students can use these concepts to prove new ideas and relationships.

FUTURE CONNECTIONS

Parallelograms and their properties show up in a number of places. A few proofs will require identifying a parallelogram and then using its properties. Later in the course we will also be classifying quadrilaterals through coordinate proof and so knowing these relationships is essential.

MY REFLECTIONS (over line l)

I rushed this area and to be honest I approached it very much like I had for the last 15 years.... I really missed an opportunity. I now see how symmetry and the transformations can reveal these properties quickly and very intuitively. Next time around I will establish many of these relationships and proofs through the tools that we have been working on from the past few objectives.

I want to develop quadrilateral properties from the triangle properties… all quadrilaterals are made up of different types of triangles.  Knowing the triangle properties and how they are rotated or reflected within the quadrilateral reveals many of its properties.