Kites and trapeziums

Join two identical triangles along a matching side and you get a quadrilateral. Which one you get depends on the triangles and on which side you join.

Two equilateral triangles of side 8 cm give a quadrilateral with all four sides 8 cm: a rhombus (with angles 60°, 120°, 60°, 120°). Two isosceles triangles with sides 8, 8 and 6 cm can be joined along an 8 cm side or along the 6 cm side, giving a different quadrilateral each time.

Now join two scalene triangles with sides 6, 9 and 12 cm along the 12 cm side. The result looks like a kite, and its neighbouring sides come in equal pairs.

Kite: a quadrilateral that can be labelled ABCD so that AB = BC and CD = DA. Notice the difference from a parallelogram: there it's the opposite sides that match; here it's the adjacent (neighbouring) ones. (A kite isn't defined by having no parallel sides: a rhombus is a kite too.)

A kite's main diagonal does a lot of work. In kite ABCD, the diagonal BD:

• bisects ∠ABC and ∠ADC;
• bisects the other diagonal, so AO = OC;
• is perpendicular to it.

All of this follows once you notice that ∆AOB ≅ ∆COB.

The last shape comes from relaxing the parallelogram. A parallelogram needs both pairs of opposite sides parallel. Ask for less: a trapezium is a quadrilateral with at least one pair of parallel opposite sides. ('At least one' means every parallelogram is a trapezium too.)

In trapezium PQRS with PQ ∥ SR, the transversal property still gives ∠S + ∠P = 180° and ∠R + ∠Q = 180°, which is enough to find the other angles from two of them. When the two non-parallel sides are equal, it's an isosceles trapezium.

In isosceles trapezium UVWX with UV ∥ XW, drop perpendiculars XY and WZ onto UV. Since XW ∥ UV, XWZY is a rectangle. Then ∆UXY ≅ ∆VWZ, so ∠U = ∠V: in an isosceles trapezium, the angles opposite the equal sides are equal (the two angles at the ends of each parallel side).

Put it all together and the chapter's Venn diagram has:

• squares inside both rhombuses and rectangles;
• those inside parallelograms;
• parallelograms inside trapeziums;
• kites overlapping the rhombuses, since a rhombus is exactly a kite that is also a parallelogram.