Lesson 5.3
Special Parallelograms: Rectangles, Rhombi, and Squares
Focus: one constraint at a time — rectangle diagonal theorem via SAS — rhombus diagonal theorem via base angles — square by inheritance
This lesson adds constraints to the parallelogram one at a time. A rectangle is a parallelogram with all four angles right. A rhombus is a parallelogram with all four sides equal. A square satisfies both conditions simultaneously. Each additional constraint produces additional diagonal properties, and both are proven rather than stated.
The rectangle proof is clean: SAS on the two triangles formed by one diagonal. The two legs adjacent to one base angle are shared side AB and the parallelogram-guaranteed equal opposite side BC = AD. The included angles are both 90° by the rectangle definition. Diagonals equal by CPCTC.
The rhombus proof is richer and explicitly invokes the base angles theorem from Unit 4. Each diagonal of a rhombus divides it into two isosceles triangles. The equal sides of the rhombus are the equal sides of the isosceles triangles, and the base angles theorem guarantees the base angles are equal. This is worth naming explicitly: ‘We just used the base angles theorem from Unit 4 to prove something about a quadrilateral. This is what it means to build a mathematical system — results from earlier proofs become tools for later ones.’
The square properties follow without new proof. A square is a rectangle, so it inherits equal diagonals. A square is a rhombus, so it inherits perpendicular diagonals and diagonal bisection of vertex angles. No new proof is needed for any of this. The most constrained shape requires the least new work. Students who find this satisfying have understood the point of the hierarchy.
Notes in this lesson use the share-and-revise format: students write their own conditional statements and proof sketches individually first, then groups share aloud, then students revise. Do not show the consolidation proofs before students have written their own versions.