Properties of a rectangle, and why the square is the special case

These teaching notes are for members

What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

Create a free account Sign in

What to assume they know

The carpenter's problem: how to be sure a frame really is rectangular — equal diagonals bisecting each other, and the isosceles triangles they create
How a property gets deduced, and why you still check it against a real shape — Deduction 4, which is what allows the definition of a rectangle to be reduced to one clause
SSS as a congruence condition, and the fact that matching parts of congruent triangles are equal
Two angles on a straight line total 180°
That a triangle with two equal sides has two equal angles facing them
Reading a Venn diagram: a region stands for a collection of objects, and a region inside another means every one of these is also one of those

What they should be able to do

State the four numbered properties of a rectangle in the chapter's order, and for each one name the argument it came from
Explain why a rectangle's opposite sides have to be parallel, using the co-interior angle test on a side acting as a transversal
Decide, from marked side lengths alone, which of a set of quadrilaterals are rectangles, and defend the answer for the one that looks different
State the definition of a square and say precisely what it adds to a rectangle
Explain why every square is a rectangle and why the reverse fails, and draw the Venn diagram that records this
Prove that the diagonals of a square cross at right angles, by finding the two triangles that SSS matches and using the straight angle at the crossing
Prove that a diagonal of a square splits each corner into two 45° halves
Construct a square from a diagonal of a given length

Where it usually goes wrong

"A square is not a rectangle — they're different shapes." The commonest error in the chapter's territory, and the reason the cartoon is on the page. Every condition in the rectangle's definition is met by a square.
"A rectangle is a square with unequal sides." Backwards. There is no requirement anywhere that a rectangle's adjacent sides differ, so the family of rectangles includes the ones where they happen to match.
"Figure (iv) has all sides equal, so it can't be a rectangle." It has four right angles, which is the whole test. Equal sides are extra, not disqualifying.
"The diagonals of any rectangle meet at right angles." They do not, and the 60° figure of the previous topic is the counter-picture. Only when the rectangle is a square does the crossing become square.
"Diagonals crossing at 90° make a square." Not on their own — they must also be equal and halve each other. Perpendicular halving diagonals of unequal length give the rhombus of §4.4.
"The two 45° halves are just what it looks like." They are forced: the triangle cut off by a diagonal has two sides of the square as its legs, so it is isosceles, and its apex is the 90° corner.
"A Venn diagram is decoration." Here it carries a claim. Drawing the square region inside the rectangle region asserts that no square escapes; drawing them overlapping instead would assert something false.

Questions to check understanding

Find every remaining angle in a rectangle given one angle between a side and a diagonal, or one angle at the crossing of the diagonals
Given side measurements only, decide which figures are rectangles and justify the decision
State whether a given claim is true or false with a reason: a quadrilateral with four equal sides and one right angle — is it a square? (Part I p.108, item 8); a quadrilateral whose diagonals are equal and halve each other — must it be a square? (Part I p.108, item 11(i))
Construct a square from its diagonal without a protractor (Part I p.108, item 6)
Identify the quadrilateral formed by two perpendicular diameters of a circle and justify it (Part I p.94, item 3)
Find the quadrilateral joining the midpoints of a square's sides, first by reasoning and then by measuring (Part I p.108, item 7)
Place a given quadrilateral correctly inside a Venn diagram of the named families

Examples worth working on the board

The four properties of a rectangle (Part I, §4.1, p.90). In the chapter's order: every angle is 90°; opposite sides match in length; opposite sides are parallel; the diagonals match in length and each cuts the other in half.
The transversal argument (Part I, §4.1, p.90). In rectangle ABCD with B top left, C top right, D bottom right, A bottom left: the side AB crosses both AD and BC, and the two angles it makes with them are 90° and 90°. Their total is 180°, which is the condition for those two lines to be parallel. The chapter leaves the second pair to the student.
A Special Rectangle — the four figures (Part I, §4.1, p.90). Four quadrilaterals with right-angle marks at every corner and these side labels: (i) 5 cm and 2 cm, drawn tilted; (ii) 3.6 cm and 6 cm, upright; (iii) 5 cm and 1 cm, drawn tilted; (iv) 4 cm on all four sides. The question printed above them asks which are not rectangles. Hand over the four sets of measurements and the question; the point of (iv) is that its answer surprises students.
The two-identities cartoon (Part I, §4.1, p.91). A person tells two curious aliens that he is Indian and also Malayali; they ask how he can be both. It is the chapter's own analogy for a square being a square and a rectangle at once, and it is worth keeping as an idea even though the artwork should be redrawn.
The two Venn diagrams (Part I, §4.1, p.91). First a single region labelled Square, with the note that every point inside stands for a square. Then a larger region labelled Rectangle with the Square region drawn wholly inside it.
The carpenter's problem, second time (Part I, §4.1, p.92). Same 8 cm diagonal, but now the thread has to make a square. The two conditions already established — equal diagonals, each halving the other — are carried over, and the open question is whether choosing the crossing angle can force the four sides to become equal.
Deduction 5, the inputs (Part I, §4.1, p.92). Square ABCD with A top left, B top right, C bottom right, D bottom left. The printed figure draws both full diagonals, A–O–C and D–O–B, meeting at O; only the segment BO carries the shared-side label. Compare the two triangles standing on it. Their other sides are the two sides of the square that meet at B, marked equal with single ticks, and the two halves of AC, marked equal with double ticks. SSS applies. Worked: the two angles at O are therefore equal, and they lie along a straight line, so each is 90°.
The 8 cm square (Part I, §4.1, p.93). Construct a square whose diagonal measures 8 cm, using only the diagonal facts. This is a construction task.
Property 5, the inputs (Part I, §4.1, p.93). Square ABCD, A top left, B top right, C bottom right, D bottom left, with the diagonal from A to C drawn and four angles marked ∠1, ∠2, ∠3, ∠4. The chapter works one triangle: inside ∆ADC, ∠1 + ∠3 + 90 = 180, and because two sides of that triangle are sides of the square, ∠1 = ∠3. Worked: each is 45°. The other two are left for the student.
Exercise inputs from Part I p.94 (§4.1, "Figure it Out"). Item 1(i): a rectangle ABCD — D top left, C top right, B bottom right, A bottom left — with both diagonals drawn, tick marks showing all four half-diagonals equal, and 30° marked at A between the side AB and the diagonal AC. Item 1(ii): a rectangle PQRS — Q top left, R top right, S bottom right, P bottom left — with both diagonals drawn, the same tick marks, and 110° marked at the crossing between the two upper half-diagonals. Item 2: diagonals both 8 cm, halving each other, at 30°, 40°, 90° and 140°. Item 3: a circle centred O with two perpendicular diameters PL and AM — identify the figure APML. Item 4: with two equal sticks and a thread and no paper, make an exact right angle. Item 5: is having opposite sides parallel and equal enough to define a rectangle?

Figures to have open

The four measured quadrilaterals of Part I p.90 with their exact labels — 5 and 2, 3.6 and 6, 5 and 1, and 4 on all sides — and right-angle marks at every corner. The tilt of (i) and (iii) is part of the exercise and should be kept.
A rectangle with one side extended slightly to read as a transversal, the two 90° marks and the parallel arrows on the other pair. Standard schematic.
The two Venn diagrams of Part I p.91 as clean schematics: one region alone, then one region nested inside a larger one, both labelled.
Square ABCD with both diagonals, the crossing point marked, the pair of triangles used in Deduction 5 shaded, and the straight angle at the crossing drawn as a single line before it is split.
Square ABCD with one diagonal only, and the four marked angles in the positions Part I p.93 uses.
The two exercise rectangles of Part I p.94, with the tick marks on the four half-diagonals and the 30° and 110° marks in the printed positions.
The two-identities cartoon should be replaced by a simple two-label graphic; do not redraw the printed artwork.

Where this sits in the book

NCERT Ganita Prakash Class 8, Part I, printed Chapter 4, "Quadrilaterals", §4.1 "Rectangles and Squares", Part I pp.90–93, covering the numbered property list, the unnumbered bold subheadings "A Special Rectangle" (Part I, p.90) and "Properties of a Square" (Part I, p.93), and Deduction 5 (Part I, p.92).
The Venn diagrams and the two-identities cartoon: Part I p.91.
The chapter-wide summary of square and rectangle properties: Part I p.109.
Exercises: Part I p.94, "Figure it Out" items 1–5; Part I p.108, items 6, 7, 8 and 11(i).