The trapezium: what a single pair of parallel sides forces, and what "isosceles" adds

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What to assume they know

The parallelogram: everything that follows from "opposite sides parallel" — the parallelogram, and the transversal facts it was built from
Properties of a rectangle, and why the square is the special case — the rectangle, which appears inside the isosceles trapezium construction
Why the angles of any quadrilateral add to 360° — the 360° total, used to check the angle work
Constructing parallel lines, and dropping a perpendicular from a point to a line
Congruence of right triangles, and reading equal sides off a figure's tick marks
That two angles on the same side of a line crossing two parallel lines total 180°

What they should be able to do

State the definition of a trapezium and explain what the words "at least" do
Explain why every parallelogram counts as a trapezium, and locate both in the chapter's summary diagram
Identify, in a trapezium, which pairs of angles must total 180° and which need not
Find the remaining angles of a trapezium given the two you are told, and say why one angle would not be enough
State what makes a trapezium isosceles
Construct an isosceles trapezium from two parallel lines and two equal slanting sides
Show that the two base angles of an isosceles trapezium are equal, by dropping two perpendiculars, naming the rectangle that appears and matching the two right triangles
Judge the claim that isosceles trapeziums are parallelograms, and justify the verdict

Where it usually goes wrong

"A trapezium has exactly one pair of parallel sides." The chapter's wording allows more, and its summary diagram draws every parallelogram inside the trapezium region. This is a genuine point of disagreement between textbooks.
"A parallelogram is not a trapezium." By this chapter's definition it is. Students who deny it will misread the summary diagram and get item 11(vii) wrong for the wrong reason.
"An isosceles trapezium is a parallelogram, because it looks symmetric." It is not, unless the slanting sides are also parallel — and then the figure is a parallelogram and the word "isosceles" has stopped doing any work. The exercise asks exactly this.
"One angle is enough to find the rest." In a parallelogram it is. In a general trapezium it is not: the two supplementary pairs are independent, so you must be told one angle from each pair. That is why the chapter's exercise figures carry two given angles, or one plus tick marks.
"The parallel sides are always horizontal, with the long one at the bottom." The second exercise figure is drawn tilted for exactly this reason.
"Base angles means the two at the bottom of the page." It means the two at the ends of one parallel side, however the figure is turned.
"The rectangle in the middle is obvious." It is argued: the two dropped segments make right angles with the lower side by construction, and the other two right angles come from the upper side being parallel to it.

Questions to check understanding

Find the remaining angles of a trapezium given two of them (Part I p.107, item 3)
Find the remaining angles of an isosceles trapezium given one angle and the equal sides
Construct a trapezium, and an isosceles trapezium, and verify the angle relations by measuring (Part I p.106)
Show that the base angles of an isosceles trapezium are equal, naming the congruence used
True or false with justification: isosceles trapeziums are parallelograms (Part I p.109, item 11(vii))
Place a given quadrilateral correctly in the chapter's whole-family diagram, and justify the placement
Decide whether a figure with one pair of parallel sides can also be a kite

Examples worth working on the board

The definition and its figure (Part I, §4.6, p.106). Trapezium PQRS drawn with P upper left, Q upper right, S lower left, R lower right; PQ is parallel to SR, the lower side is the longer of the two, and arcs mark the two angles at S and at R. The chapter asks the student to construct one and measure those two marked angles.
Property 1 (Part I, §4.6, p.106). Because PQ is parallel to SR, the angle at S and the angle at P total 180°, and so do the angle at R and the angle at Q. The chapter then says the remaining angles follow and asks the student to check by measuring.
The isosceles condition (Part I, §4.6, p.106). When the two sides that are not the parallel pair have the same length, the trapezium is called isosceles.
The construction (Part I, §4.6, p.106). Isosceles trapezium UVWX with UV parallel to XW, drawn with X upper left, W upper right, U lower left, V lower right. Two stages are printed: first two parallel lines with the four letters placed and the instruction to make UX and VW the same length, then the closed figure. The task attached is to measure the angle at U.
The perpendicular trick, the inputs (Part I, §4.6, p.106). Drop XY and WZ perpendicular to UV, with Y and Z the feet. Because XW is parallel to UV, the two angles marked a and b at X and at W each come to 90°, so the figure in the middle has four right angles — the chapter states it is a rectangle.
The congruence (Part I, §4.6, p.107). The two right triangles at the ends, the one on UX and the one on VW, are asserted to be congruent and the reason is left to the student. From that, the angles at U and at V are equal, which the chapter records as Property 2. An added reading, for the student to supply: the two slanting sides are equal by assumption, the two perpendiculars are equal because they are opposite sides of the rectangle, and both triangles carry a right angle.
Exercise item 3 (Part I, §4.6 exercise, p.107). Two trapeziums, angles to be completed. The first is drawn with a long upper side and a short lower side, both carrying arrow marks to show they are the parallel pair, and the two angles at the ends of the short side are given as 135° and 105°. The second is drawn tilted: the parallel pair is marked with arrows on two of the sides, the other two sides carry a tick mark each — so it is isosceles — and one angle is given as 100°.
Exercise item 4 (Part I, §4.6 exercise, p.107). One Venn diagram covering five families — parallelogram, kite, rhombus, rectangle, square — followed by three questions about where the kite fits among them. The trapezium is not in that list; it enters in the chapter's own summary diagram instead.
Exercise item 11(vii) (Part I, p.109). True or false, with justification: isosceles trapeziums are parallelograms. Input only.
The whole-family diagram (Part I, p.110). The summary places Parallelogram entirely inside Trapezium, Rectangle inside Parallelogram, Kite overlapping Parallelogram with the overlap named Rhombus, and Square in the overlap of Rectangle and Rhombus. The Kite region reaches outside the Trapezium region.

Figures to have open

Trapezium PQRS in the printed orientation with the parallel pair marked and the two base angles arced (Part I p.106).
The two-stage construction of the isosceles trapezium UVWX (Part I p.106), with the equal-length instruction visible in the first stage.
The same trapezium with XY and WZ dropped to UV, the feet Y and Z labelled, the middle rectangle picked out, and the two end triangles shaded separately.
The two exercise trapeziums of Part I p.107, item 3 — the first upright with 135° and 105° at the ends of the shorter parallel side, the second tilted with 100° and a tick on each of the two non-parallel sides. The arrow marks showing which pair is parallel must be kept; without them the figures are ambiguous.
The chapter's whole-family Venn diagram from Part I p.110, redrawn with all six labels and with the kite region reaching outside the trapezium region.

Where this sits in the book

NCERT Ganita Prakash Class 8, Part I, printed Chapter 4, "Quadrilaterals", §4.6 "Kite and Trapezium", unnumbered bold subheading "Trapezium", Part I pp.106–107. The definition, Property 1, the isosceles construction and the perpendicular argument occupy Part I p.106; the congruence and Property 2 finish at the top of Part I p.107.
Exercises: Part I p.107, items 3 and 4; Part I p.109, item 11(vii).
The definitions of trapezium and kite restated, and the whole-family Venn diagram: Part I p.110.