Squares

Look at four quadrilaterals with all angles 90°, one of which has all four sides 4 cm. Is that last one a rectangle? Yes: it fits the definition exactly. It's just a special rectangle, the one with all sides equal, and we call it a square.

Square: a quadrilateral in which all the angles are 90° and all the sides are equal. So every square is a rectangle, while clearly not every rectangle is a square.

The book makes the point with a joke: 'I am an Indian and I am a Malayali.' 'Wait, which one of them are you?' You're both, because one group sits inside the other.

A Venn diagram shows this. A set of things is drawn as the points inside a closed curve, usually an oval. Draw the oval of all squares entirely inside the oval of all rectangles, and the relationship is on the page.

Back to the carpenter. Equal diagonals bisecting each other give a rectangle at any angle. What extra condition makes the sides equal too?

Deduction 5. Suppose the result is a square ABCD. Compare ∆BOA and ∆BOC:

• BO is in both;
• OA = OC, since the diagonals bisect each other;
• BA = BC, since all sides of a square are equal.

Three sides, so by SSS, ∆BOA ≅ ∆BOC.

So ∠BOA = ∠BOC, as matching parts. But together they make a straight angle: ∠BOA + ∠BOC = 180°. So each must be 90°.

So the diagonals of a square bisect each other at right angles, and since the ends of the diagonals fix the corners, drawing them that way gives a square. (Two diameters of a circle at right angles, for example, have their ends on the four corners of a square.)

The properties of a square:

• all sides equal;
• opposite sides parallel;
• all angles 90°;
• diagonals equal, bisecting each other at 90°;
• the diagonals bisect the angles of the square: in ∆ADC, AD = DC, so ∠1 = ∠3, and ∠1 + ∠3 + 90 = 180 gives 45° each.

The square has every property of a rectangle, and (as you'll see) every property of a rhombus too.