Concepts: Quadrilaterals (part 1)

1.1 What is a quadrilateral

A quadrilateral is a closed figure made of four line segments. The segments are its sides, the points where they meet are its vertices, and the openings between two sides at a vertex are its angles. A segment joining two non-adjacent vertices is a diagonal, and every quadrilateral has two diagonals.

The textbook opens with five figures. The first three are quadrilaterals; the other two are not, because their sides cross or the figure does not close. A quadrilateral must be a simple closed polygon: four sides, four vertices, four interior angles, and no crossing sides.

We name a quadrilateral by listing its vertices in order around the figure. Quadrilateral ABCD has sides AB, BC, CD and DA, and its angles are ∠A, ∠B, ∠C and ∠D. The order matters: naming the same four points in a different order may describe a different figure or a self-crossing one.

1.2 The angle sum is 360 degrees

Draw diagonal AC in quadrilateral ABCD. It splits the figure into two triangles, △ABC and △ACD. Each triangle has an angle sum of 180 degrees. The six angles of the two triangles together make up the four angles of the quadrilateral, so the total is 180 + 180 = 360 degrees.

This single result explains several facts. A quadrilateral cannot have three right angles and a fourth angle that is not a right angle, because the fourth would have to be 360 - 270 = 90 degrees. It also lets us find a missing angle whenever the other three are known.

Worked example 1. Three angles of a quadrilateral measure 85°, 95° and 100°. Find the fourth angle. The sum of the three angles is 85 + 95 + 100 = 280°. The fourth angle is 360 - 280 = 80°. Check: 85 + 95 + 100 + 80 = 360. The answer is 80°.

1.3 Rectangles

A rectangle is a quadrilateral in which all four angles are right angles and the opposite sides are equal. The textbook shows that this definition can be shortened: if all four angles are 90°, the opposite sides are forced to be equal, so a rectangle can also be defined as a quadrilateral with all angles 90°.

A second definition comes from the carpenter problem. A carpenter has one 8 cm strip and wants the endpoints of two strips, tied with a thread, to form a rectangle. If the second strip is also 8 cm long and the two strips are joined at their midpoints, the endpoints form a rectangle. So the strip construction gives exactly this test: if a four-sided figure has matching diagonals and each one cuts the other into two equal halves, the figure is a rectangle.

The two definitions describe the same family. A rectangle has four right angles, both pairs of opposite sides equal and parallel, and two diagonals of equal length that cut each other in half. Each property can be proved from the others by congruence, which is the work of the next page.

1.4 Worked example: angles in a rectangle

Worked example 2. In rectangle ABCD, the diagonal BD makes an angle of 30° with AB. Find the other angles marked in the figure. Triangle ABD is right angled at A, so ∠ADB = 180 - 90 - 30 = 60°. In triangle AOB, where O is the intersection of the diagonals, OA = OB because the diagonals of a rectangle are equal and bisect each other. So △AOB is isosceles and ∠OAB = ∠OBA = 30°. Then ∠CAD = 90 - 30 = 60°. In triangle ACD, ∠CAD = 60° and ∠ADC = 90°, so ∠ACD = 30°. In triangle ABC, ∠CAB = 30° and ∠ABC = 90°, so ∠ACB = 60°. Also ∠BDC = 90 - 60 = 30°.

1.5 Squares

A square is a quadrilateral with all angles 90° and all sides equal. Every square is a rectangle and also a rhombus, because it satisfies both definitions. The reverse is not true: a rectangle can have unequal adjacent sides, and a rhombus can have angles other than 90°.

The diagonals of a square have a strong property. They are equal (as in every rectangle), they bisect each other, they meet at 90°, and each diagonal bisects the angles at its ends. In triangle ADC, AD = DC and ∠ADC = 90°, so the two base angles are equal: ∠DAC = ∠ACD = 45°. The same argument at the other vertices shows that every diagonal divides a square's right angle into two 45° angles.

1.6 Parallelograms

A parallelogram is a quadrilateral whose opposite sides are parallel. Every rectangle is a parallelogram, and every rhombus is a parallelogram, but a parallelogram need not have right angles or equal adjacent sides.

The properties of a parallelogram follow from its parallel sides. Adjacent angles lie on the same side of a transversal, so they add to 180°. Opposite angles are equal, opposite sides are equal, and the diagonals bisect each other. These are proved with the congruence of the two triangles formed by a diagonal.

Worked example 3. In parallelogram ABCD, ∠A = 30°, AB = 4 cm and AD = 5 cm. Find the other angles and sides. Adjacent angles add to 180°, so ∠B = 150° and ∠D = 150°. Opposite angles are equal, so ∠C = 30°. Opposite sides are equal, so CD = 4 cm and BC = 5 cm.

1.7 The family of quadrilaterals

The definitions fit inside one another. Every rectangle, rhombus and square is a parallelogram. A square is the overlap of a rectangle and a rhombus: it has four right angles and four equal sides. A kite has two pairs of adjacent equal sides, and a trapezium has at least one pair of parallel sides. A quadrilateral can belong to more than one family, which is why the classification is shown with a Venn diagram rather than a list.

The next page proves the properties in full, including the deductions about diagonals that the textbook uses to define rectangles and squares. The exercise pages then work through every Figure it Out question in the chapter.