Why a rectangle's diagonals are equal, and when they split the corners evenly

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What to assume they know

The two properties that define a rectangle, and the one more a square needs — the rectangle and square conditions
Naming corners in order, and why a rotated square is still a square — corners named in order, so that PR and QS name something definite
Reading and drawing an angle with a protractor — measuring an angle with a protractor
The shortest and longest crossing inside a rectangle — AC and BD already met as the two longest crossings

What they should be able to do

Draw both diagonals of a rectangle and name them correctly
Identify which pairs of angles the chapter calls opposite angles
Predict whether the two diagonals are equal, then check by measuring
Say which of the eight angles made by the diagonals are equal to which
Explain why the two parts of one corner must add to 90°
Determine the condition under which a diagonal splits its corners into equal halves
Set up a record of an experiment: name the quantities to be tracked before collecting any
Distinguish a pattern confirmed by several trials from one that has been explained

Where it usually goes wrong

"A diagonal cuts the corner in half." It does not, except in a square. This is the single most common wrong belief on this material, and the chapter's own two questions on p.204 are aimed straight at it.
"The diagonals are equal because they look equal." They are equal, but looking is not why. The chapter has students predict and then measure — and then asks how anyone could be sure it always holds. Treat that last question as the point of the section, not as a rhetorical flourish.
"Eight angles means eight numbers to find." Two numbers. The other six are copies. Getting students to see that is worth more than any single measurement.
"The two diagonals cross at right angles." In a square they do; in a general rectangle they do not. Nothing in this section says they do.
"The 45°/45° case is a coincidence." It is the definition of the special case. Equal halves at a corner means the two sides at that corner are equal, which is what makes the rectangle a square.
"Measuring more rectangles will eventually prove it." No number of trials proves a general law, which is precisely what the chapter asks at the end of p.204.

Questions to check understanding

Draw both diagonals of a rectangle and measure them. What do you find?
Are the two parts a diagonal makes at one corner equal? (asked at p.204)
Identify all the pairs of equal angles in the figure (asked at p.204)
What must be true of a rectangle for a diagonal to halve the corners it runs between? (asked at p.204)
Construct a rectangle whose diagonal makes 50° and 40° at the corners it runs between (asked at p.211)
Construct one where the division is 45° and 45°, and say what happens to the sides (asked at p.211)
One part of a corner measures 35°. What does the other part measure, and why?

Examples worth working on the board

The figure on p.203. A rectangle PQRS with P at the top left, Q at the top right, R at the bottom right and S at the bottom left. Both diagonals are drawn: PR and QS. Eight small angles are marked with lowercase letters. Checked against p.203, since every one of those letters is artwork. In the printed figure the letters sit like this — at P, d against the top side and c against the left side; at Q, e against the top side and f against the right side; at S, b against the left side and a against the bottom; at R, g against the right side and h against the bottom.
What the eight angles actually come to. Group them by which side of the rectangle they lean on. The four leaning on the two long horizontal sides — d, e, a and h — are all equal to one another. The four leaning on the two vertical sides — c, f, b and g — are all equal to one another. One from each group makes up a whole corner, so one of the first kind plus one of the second comes to 90°.
A worked instance. Take a rectangle 7 cm across and 4 cm high, the same one §8.4 explored. Measuring the angle a diagonal makes with the long side gives close to 30°, and with the short side close to 60°; the two come to 90°, and every one of the eight angles reads as one or the other. Any rectangle whose sides are in a different proportion gives a different pair — but always just one pair.
The two questions the chapter asks outright, p.204. Whether g matches h; and separately, whether c matches d. Each asks about the two halves of one corner, and for a general rectangle the honest answer is no. They match only when the rectangle is a square.
The Explore, p.204. What must be true of the rectangle for a diagonal to halve the corners it runs between? The two sides meeting at such a corner have to be equal — that is, the rectangle has to be a square. The chapter poses this and leaves it to experiment.
The recording table, p.204. A printed table with a column headed Sides and eight further columns headed A to H, with two blank rows underneath. Checked against p.204. The point of it is that you name what you will track before you collect anything, and the chapter says outright which quantities those are: the sides, and the eight angles the diagonals make.
Evidence from the constructions. The Construct list at p.211 asks for a rectangle in which a diagonal makes 50° and 40° at the corners it runs between, and then one where the split is 45° and 45°, with a follow-up question about what the sides do. The second is the square, arrived at from the other direction. Use it as the confirmation of section 10.

Figures to have open

PQRS with both diagonals and all eight lettered angles, drawn faithfully to the printed arrangement. The letters' positions matter to everything in this topic, so this figure must follow the printed page.
A colour-coded version of the same figure with the two groups of four distinguished.
The dragging rectangle for section 9, going through the square.
The recording table with a Sides column and eight angle columns.
No photograph or dataset is required.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 8 "Playing with Constructions", §8.5 "Exploring Diagonals of Rectangles and Squares", pp.203–204 — the naming of the diagonals and the lettered figure on p.203, and the opposite-angles passage, the two questions, the Explore, the recording table, the square as special case and the closing Math Talk on p.204
§8.5, p.211, Construct items 1 and 2, for the 50°/40° and 45°/45° cases
§8.4, p.199, where AC and BD are compared without yet being called diagonals
§8.2, p.193, for the conditions that make the square the special case