Why the angles of any quadrilateral add to 360°

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What to assume they know

That the three angles of a triangle total 180°
What a diagonal is, and that a quadrilateral has two of them
Properties of a rectangle, and why the square is the special case — the properties of a rectangle, which supply the puzzle this section opens with
Adding and regrouping angle measures, and reading numbered angles off a figure
Constructing a quadrilateral from given angles, well enough to discover that one particular attempt refuses to close

What they should be able to do

State the question the section opens with: can a quadrilateral have three right angles and a fourth angle that is not a right angle?
Split a quadrilateral into two triangles with one diagonal, and identify which triangle angle belongs to which corner of the quadrilateral
Show that the six triangle angles regroup into the quadrilateral's four, and conclude that they total 360°
Use the total to explain why three right angles force the fourth
Apply the total to find a missing angle in a quadrilateral given the other three
Examine a quadrilateral that caves inwards, say which of its two diagonals can be used, and state which angle at the caved-in corner has to be counted
Recognise the 360° total as a reused fact rather than a new one

Where it usually goes wrong

"360° is a separate rule you have to memorise." It is 180° used twice. A student who sees the diagonal never has to remember the number.
"Any diagonal will do." For a figure that caves inwards, one of the two diagonals leaves the figure entirely, and the two-triangle argument fails with it. Which diagonal you may use is part of the reasoning, not a detail.
"The angles of a triangle add to 180°, so a quadrilateral has four triangles in it." Students over-count by drawing both diagonals, getting four triangles and 720°.
"The angle at a caved-in corner is the small one." From the inside of the figure it is the large one. Measuring the small angle and adding gives a total that is not 360°, and students conclude the rule has broken.
"You cannot draw a quadrilateral with three right angles at all." You can — the fourth then has to be 90° as well, and the figure is a rectangle. What is impossible is three right angles with a fourth angle that differs.
"Any four-sided-looking figure is a quadrilateral." The opening page shows two that are not, both because a side is curved. Four corners is not the test; four straight sides is.

Questions to check understanding

Find the fourth angle of a quadrilateral given the other three
Decide whether a stated set of four angles can be the angles of a quadrilateral
Justify why a quadrilateral cannot have exactly three right angles
True or false with justification: does having three square corners force a quadrilateral to be a rectangle (Part I p.109, item 11(ii))?
Show that the total is still 360° for a quadrilateral that caves inwards, by reasoning and by measuring (Part I p.108, item 10)
Given a quadrilateral figure, say whether it is one at all, and why

Examples worth working on the board

The opening question (Part I, §4.2, p.94). Can a quadrilateral be built with three of its angles equal to 90° and the fourth not 90°? The chapter reports that students trying constructions find it does not seem to come out, and then asks why.
**Quadrilateral *SOME*** (Part I, §4.2, p.95). This is the chapter's own worked figure and its labelling should be kept, because the regrouping step depends on it. The corners: S upper left, O upper right, M lower right, E lower left. The diagonal drawn is SM, splitting the figure into the triangle containing E and the triangle containing O. Six angles are numbered: at S, ∠1 lies in the lower triangle and ∠4 in the upper; at M, ∠3 lies in the lower triangle and ∠6 in the upper; ∠2 sits alone at E and ∠5 alone at O.
The two triangle totals. In the lower triangle, ∠1 + ∠2 + ∠3 = 180°. In the upper, ∠4 + ∠5 + ∠6 = 180°. Adding gives 360° across the six.
The regrouping. ∠1 with ∠4 is the whole corner at S; ∠3 with ∠6 is the whole corner at M; ∠2 and ∠5 are corners already. So the four corner angles of the quadrilateral total 360°.
The three-right-angle case. Worked: 90 + 90 + 90 = 270, leaving 90 for the fourth. There is no room for it to be anything else, which is why the construction refuses to close.
The caved-in figure (Part I, §4.6 exercise, p.108, item 10). A quadrilateral drawn as B at the top, A at the lower left, C at the lower right, and D sitting between them and above the line AC — inside the triangle ABC. Angles are marked at all four corners, including at D. The exercise asks whether the 360° total still holds and wants both a reasoned answer and a measured one. Inputs only. Two things: the segment BD stays inside the figure while the segment AC does not, and the angle at D that belongs to the inside of this quadrilateral is the larger of the two angles there.
The chapter's own caved-in quadrilateral (Part I, p.82, figure (iii)). On the opening page the chapter shows five figures and states that (i), (ii) and (iii) are quadrilaterals while the other two are not. Figure (iii) is an arrowhead-shaped four-sided figure, and its angles are marked like the others. Figure (iv) has one side drawn as a curve and figure (v) has two, rather than straight segments, which is why both are excluded; the chapter poses this as a question and does not print the reason.
Where the total gets spent later. In §4.3 the parallelogram's angles are found from supplementary pairs and the total checks them; in §4.6 the trapezium's two remaining angles are found the same way. Both are inside this chapter (Part I, pp.97 and 106).

Figures to have open

Quadrilateral SOME with the diagonal SM and all six numbered angles in the printed positions (Part I p.95). The labelling is load-bearing for section 5; redraw as a schematic but keep the letters and numbers where the chapter puts them.
The same figure shown step by step so that ∠1 and ∠4 visibly merge into one corner angle. This is the single most important frame in the explanation.
A quadrilateral with both diagonals drawn, showing four triangles, used only to expose the double count in the misconception section.
The caved-in quadrilateral of Part I p.108, item 10, with D inside triangle ABC, both candidate diagonals drawn, and the two angles at D distinguishable.
The five opening figures of Part I p.82 as clean line drawings, with (iv) drawn with one curved side and (v) with two. Redraw rather than reproduce.

Where this sits in the book

NCERT Ganita Prakash Class 8, Part I, printed Chapter 4, "Quadrilaterals", §4.2 "Angles in a Quadrilateral", Part I pp.94–95. The section is short — it starts at the foot of Part I p.94 and finishes in the upper half of Part I p.95.
Chapter opening page, Part I p.82: the five figures and the statement of which three are quadrilaterals.
Exercise using the result on a caved-in figure: Part I p.108, item 10, flagged in the margin as a Try This.
The result restated in the chapter summary: Part I p.110.
Later uses of the total inside the chapter: Part I pp.97 and 106.