Quadrilaterals
This chapter studies figures bounded by four straight sides. You already know the rectangle and the square; here you meet their relatives, the parallelogram, the rhombus, the kite and the trapezium. Our entry point is a frame-maker who wants to cross two metal rods so that a wire stretched around their tips snaps into a perfect rectangle. Her puzzle reveals a quiet truth: the diagonals of a four-sided figure quietly decide its whole shape. Working from that idea, we will prove, not merely measure, that the angles of any quadrilateral total 360^\circ, that a parallelogram’s opposite sides are forced to be equal, and that a rhombus’s diagonals must cross at right angles. The engine behind every proof is triangle congruence, and by the close of the chapter you will read geometry as a sequence of justified steps, where each result is argued into existence rather than merely noticed.
Learning Outcomes
By the end of this chapter, you will be able to:
- recognise and define the principal quadrilaterals, rectangle, square, parallelogram, rhombus, kite and trapezium, and list the properties of each;
- apply triangle congruence (SAS, ASA, AAS, SSS) to justify claims about sides, angles and diagonals;
- explain why every quadrilateral has an angle sum of 360^\circ;
- describe how the diagonals, their lengths, where they meet, and the angle between them, pin down which quadrilateral results;
- organise these shapes in a Venn diagram showing that each square is a rectangle, each rectangle a parallelogram, and so on;
- compute unknown angles in rectangles, parallelograms, rhombuses, kites and trapeziums by reasoning.