4.6 Kite and Trapezium

Gluing two triangles edge-to-edge produces yet more quadrilaterals. Two of them earn special names.

The Kite

Definition: Kite A kite is a quadrilateral ABCD in which two adjacent pairs of sides are equal: AB = BC and CD = DA.

Its main diagonal BD carries a pleasing symmetry. Comparing \triangle ABD and \triangle CBD (AB = CB, AD = CD, BD shared, SSS) shows that BD bisects both \angle ABC and \angle ADC. From this, \triangle AOB \cong \triangle COB, so BD also bisects the other diagonal AC (AO = OC) and is perpendicular to it.

Kite ABCD with AB equal to BC and CD equal to DA; the axis BD bisects the angles at B and D and cuts diagonal AC at right angles at O A B C D O
Figure 4.11 · The kite: axis BD bisects ∠B and ∠D, bisects the diagonal AC at O, and crosses it at 90°.

Properties of a Kite - Two pairs of adjacent sides are equal (AB = BC, CD = DA). - One diagonal (the axis of symmetry) bisects the two angles it passes through. - That diagonal bisects the other diagonal and is perpendicular to it.

Note the link to the rhombus: a rhombus has all four sides equal, so every rhombus is a kite, but a kite (only two adjacent pairs equal) need not be a rhombus.

The Trapezium

Definition: Trapezium A trapezium is a quadrilateral with at least one pair of parallel opposite sides.

If PQ \parallel SR, then PS and QR act as transversals, so the co-interior angles satisfy \angle S + \angle P = 180^\circ, \qquad \angle R + \angle Q = 180^\circ. That lets you recover the remaining angles from any one of them.

When the two non-parallel sides are equal in length, the figure is an isosceles trapezium. Drop perpendiculars from the ends of the shorter parallel side onto the longer one; the central piece is a rectangle, and the two right triangles at the ends are congruent, which proves the base angles are equal.

Isosceles trapezium PQRS with PQ parallel to SR; perpendiculars from P and Q create a central rectangle and two congruent right triangles, so the base angles are equal P Q R S congruent corner triangles ⇒ equal base angles
Figure 4.12 · Isosceles trapezium PQRS: dropped perpendiculars leave a central rectangle and two congruent right triangles, so the base angles are equal.

Properties of a Trapezium - At least one pair of opposite sides is parallel. - Each pair of co-interior angles (between the parallel sides) adds to 180^\circ. - In an isosceles trapezium, the angles at the ends of each parallel side are equal.

The Family Tree of Quadrilaterals Drawn as nested sets:

  • Every square is both a rectangle and a rhombus.
  • Every rectangle and every rhombus is a parallelogram.
  • Every rhombus is a kite, and every parallelogram is a trapezium (its opposite sides are parallel).
  • A kite and a trapezium are the broadest families here; the square sits dead centre, belonging to every family at once.
Family tree of quadrilaterals: square inside rectangle and rhombus, both inside parallelogram, inside trapezium; rhombus is also a kite; everything is a quadrilateral Quadrilateral Trapezium Kite Parallelogram Rectangle Rhombus Square every arrow means “is a”
Figure 4.13 · The family tree of quadrilaterals — every arrow reads “is a”.
Classify the Quadrilateral

Enrichment: The Midpoint Quadrilateral Here is a surprising result worth carrying with you. Take any quadrilateral and join the midpoints of its four sides in order. The figure you get is always a parallelogram.

The reason rests on a single fact about triangles: a segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length (the midpoint theorem). Draw a diagonal of the original quadrilateral. In each of the two triangles it creates, the midpoint segment is parallel to that diagonal. So the two opposite sides of the midpoint figure are both parallel to the same diagonal, hence parallel to each other. Repeating with the other diagonal handles the second pair. Both pairs of opposite sides are parallel, so the midpoint quadrilateral is a parallelogram.

Figure it Out: Kites, Trapeziums & Mixed

Practice

  1. Find all the sides and angles of the quadrilateral formed by joining two equilateral triangles of side 5\ \text{cm} along a common edge.
  2. Construct a kite whose diagonals are 7\ \text{cm} and 9\ \text{cm}.
  3. Find the remaining angles in these trapeziums: (i) one with base angles 125^\circ and 95^\circ (with PQ \parallel SR); (ii) an isosceles trapezium with equal slant sides and one marked angle of 105^\circ.
  4. Draw a Venn diagram of parallelograms, kites, rhombuses, rectangles and squares, then answer: (i) which quadrilateral is both a kite and a parallelogram? (ii) Can a quadrilateral be both a kite and a rectangle? (iii) Is every kite a rhombus?
  5. PAIR and RODS are two rectangles arranged so that a 35^\circ angle is marked at R. Find \angle IOD.
  6. Construct a square with diagonal 7\ \text{cm} without using a protractor.
  7. ABCD is a square; U, V, W, X are the midpoints of its sides. What kind of quadrilateral is UVWX?
  8. A quadrilateral has four equal sides and one angle of 90^\circ. Must it be a square?
  9. What quadrilateral has opposite sides equal? Justify it with a diagonal and congruent triangles.
  10. Does the angle sum of a non-convex (dart-shaped) quadrilateral also equal 360^\circ?
  11. True or false, with reasons:
    1. A quadrilateral whose diagonals are equal and bisect each other must be a square.
    2. A quadrilateral with three right angles must be a rectangle.
    3. A quadrilateral whose diagonals bisect each other must be a parallelogram.
    4. A quadrilateral whose diagonals are perpendicular must be a rhombus.
    5. A quadrilateral with equal opposite angles must be a parallelogram.
    6. A quadrilateral in which all angles are equal is a rectangle.
    7. Every isosceles trapezium is a parallelogram.
  1. Two equilateral triangles (each angle 60^\circ) joined along a side give a rhombus of side 5\ \text{cm}, with angles 60^\circ, 120^\circ, 60^\circ, 120^\circ (the two shared corners add 60^\circ + 60^\circ = 120^\circ).
  2. A kite’s diagonals are perpendicular. Draw the 9\ \text{cm} diagonal AC, raise its perpendicular bisector at the midpoint T, and on that perpendicular mark B and D so that the second diagonal BD = 7\ \text{cm}. Join AB, BC, CD, DA: ABCD is the kite.
  3. (i) With PQ \parallel SR: the angle co-interior to 125^\circ is 180 - 125 = 55^\circ, and the one co-interior to 95^\circ is 180 - 95 = 85^\circ. (ii) The angle adjacent to the 105^\circ angle (along a slant side) is 180 - 105 = 75^\circ; by the isosceles symmetry the two angles on the long parallel side are each 75^\circ and the two on the short side each 105^\circ.
  4. (i) A quadrilateral that is both a kite and a parallelogram is a rhombus (with the square as a special case). (ii) No, a kite-and-rectangle would need all four sides equal and all angles 90^\circ, i.e. a square; a non-square rectangle is not a kite. (iii) No, every rhombus is a kite, but a kite need not be a rhombus.
  5. \angle IOD = 35^\circ.
  6. Draw a segment AB = 7\ \text{cm}, construct the perpendicular bisector through its midpoint O, and mark C, D on it with OC = OD = 3.5\ \text{cm}. Join AC, BC, BD, AD: since the diagonals are equal, bisect each other and meet at 90^\circ, ACBD is a square.
  7. UVWX is a square. Each side equals \tfrac{1}{\sqrt 2} of the original side (from UV^2 = \tfrac{x^2}{4} + \tfrac{x^2}{4} = \tfrac{x^2}{2}), so all four sides are equal, and each corner is 90^\circ (two 45^\circ base angles taken out of a straight angle leave 90^\circ).
  8. Yes. Four equal sides make a rhombus; in a rhombus adjacent angles are supplementary and opposite angles equal, so if one angle is 90^\circ all four are 90^\circ, a square.
  9. It is a parallelogram. Drawing a diagonal of ABCD gives \triangle ABC \cong \triangle CDA by SSS (AB = CD, BC = DA, AC common); equal alternate angles then make AB \parallel CD and BC \parallel AD, so both pairs of opposite sides are parallel.
  10. Yes, 360^\circ. A diagonal still cuts the figure into two triangles whose angles total 180^\circ + 180^\circ = 360^\circ.
    1. False, equal, bisecting diagonals give a rectangle; a square also needs them perpendicular. (ii) True, the fourth angle is forced to 90^\circ since all four total 360^\circ. (iii) True, bisecting diagonals is exactly the parallelogram condition (see the converse in Section 4.3). (iv) False, it could be a kite. (v) True. (vi) True. (vii) False, the slant sides of an isosceles trapezium are not parallel, so it is not a parallelogram.