Trapezoids and Kites

This lesson covers the quadrilaterals outside the parallelogram family. The hidden triangle strategy from Lesson 5.2 still applies here — both proofs create triangles through auxiliary constructions and use triangle congruence to extract the results. The isosceles trapezoid proof drops altitudes from the two top vertices, creating right triangles for HL. The kite proof uses the main diagonal as a shared side, applying SSS to the large triangles first and SAS to the smaller intersection triangles second.

 

Symmetry is genuinely present in both shapes and worth pointing out — the isosceles trapezoid has a vertical line of symmetry through the midpoints of both bases, the kite has its main diagonal as an axis of symmetry. Noticing the symmetry helps students anticipate which parts will be equal before they write the proof. But the proofs themselves are triangle congruence arguments. The symmetry observation motivates the approach; the congruence argument establishes the result.

 

The isosceles trapezoid base angles proof uses HL, which is established as a congruence shortcut in Lesson 3.2. The auxiliary altitude construction creates two right triangles at E and F on the base. The hypotenuses are the equal legs of the trapezoid (given). The legs of the right triangles, AE and BF, need a brief justification: both altitudes are perpendicular to the same line CD, so they are parallel to each other. Combined with AB ∥ CD, this creates a parallelogram ABFE — and since the angles at E and F are both 90°, it is a rectangle. In a rectangle opposite sides are equal, so AE = BF. Coach this step carefully — it is the one students most often cannot close on their own.

 

The kite proof is efficient once students see the axis. Two properties emerge: perpendicular diagonals (two-step SSS then SAS, ending with a linear pair of congruent angles forced to 90°) and one pair of congruent opposite angles (CPCTC from the initial SSS congruence).

 

Notes format: students carry their notes sheet to the boards from the start of the lesson. When a group’s proof is checked and approved, the teacher gives the go-ahead to record that result before moving on. This keeps notes authentic without a separate sharing session at the end of class.