Parallelogram Properties: The Engine of the Unit

Parallelograms are the logical backbone of this unit. Every rectangle, rhombus, and square is a parallelogram — so every property proven here is automatically inherited by those shapes. Students who understood the hierarchy from Lesson 5.1 will recognize this without being told.

 

Before class, draw a parallelogram ABCD with both diagonals on each group’s board. The task asks groups to prove everything they can. The drawing of both diagonals is deliberate: it makes the triangles immediately visible and invites the hidden triangle strategy before it has been named. The definition gives students two pairs of parallel sides to start from. Everything else must be proven.

 

Before the proofs begin in consolidation, name the strategy explicitly: every proof in this lesson uses the same template. Draw a diagonal to create two triangles, prove the triangles congruent, extract what you need by CPCTC. While the parallelogram is the object being studied it is the triangles inside it that light the path to discovery. This is the big idea of the unit in action. Say it, write it, and return to it whenever students lose the thread.

 

One property — opposite sides congruent — was proven in Unit 3. Students should be able to reproduce that proof here. Treat it as review and move quickly to the properties that are new.

 

The converse work — if both pairs of opposite sides are congruent, is the quadrilateral necessarily a parallelogram? — is included as an extension. Students need this converse for coordinate proof work in Lesson 5.5. The proof follows the same hidden triangle template: SSS gives congruence, CPCTC gives alternate interior angles, the angle relationship gives parallel lines.