The parallelogram: everything that follows from "opposite sides parallel"

What to assume they know

• Properties of a rectangle, and why the square is the special case — the rectangle's four properties, which the parallelogram will generalise
• Why the angles of any quadrilateral add to 360° — the 360° total, used to check the angle work
• Angles made by a transversal cutting two parallel lines: alternate ones are equal, and the two it makes on one side of itself total 180°
• The congruence conditions AAS and ASA
• Constructing a line parallel to a given line through a given point, with a ruler and a set-square or with a compass
• Reading a figure in which parallel sides are marked with arrows

What they should be able to do

• Construct a parallelogram from two adjacent sides and the angle between them
• State the definition of a parallelogram and identify a rectangle as one of them
• Prove that adjacent angles of a parallelogram total 180°, naming the transversal and the pair of angles used
• Prove that opposite angles are equal, doing the work with a letter rather than with a number so the result covers every parallelogram
• Prove that opposite sides are equal, by drawing a diagonal and matching two triangles by AAS
• Prove that the diagonals cut each other in half, by matching two triangles at the crossing by ASA
• State what is not true of a parallelogram's diagonals, and connect that to the extra condition a rectangle carries
• Complete a partially labelled parallelogram — all four sides and all four angles — from two sides and one angle

Where it usually goes wrong

• "A parallelogram is a pushed-over rectangle." That picture makes students expect equal diagonals to survive the push. They do not. Direction is preserved by the definition; length relations between diagonals are not.
• "Opposite angles are equal because the figure looks symmetric." The chapter refuses to accept that: it does the x version specifically so the claim does not rest on one drawing.
• "Adjacent angles are equal." They total 180°, which makes them equal only in the rectangle. Students who conflate the two rules get 90° everywhere.
• "The diagonals of a parallelogram are equal." The single most common error in this section, and the chapter blocks it with a measurement. Draw a long thin parallelogram and the two diagonals are obviously different.
• "Equal opposite sides is part of the definition." It is a consequence. The definition names direction only.
• "A rectangle is not a parallelogram, because it has right angles." Extra conditions never remove membership. The chapter's nested Venn diagram is on Part I p.96 for exactly this.
• "You need both diagonals for Deduction 7." One is enough, and using both at once is how students end up asserting what they are trying to prove.

Questions to check understanding

• Find the remaining angles of a parallelogram given one angle — the form of Part I p.102, item 1(i), a parallelogram with 40° at one corner, and item 1(ii), with 110° at one corner
• Find the remaining sides and angles of a parallelogram given two adjacent sides and the angle between them
• Construct a parallelogram from its diagonals: lengths 7 cm and 5 cm crossing at 140° (Part I p.102, item 2)
• True or false with justification: does each diagonal halving the other force a parallelogram (Part I p.109, item 11(iii)), and does having the two pairs of facing angles equal force one (item 11(v))?
• Say what kind of quadrilateral has equal opposite sides, and justify it, with the hint that a diagonal and a pair of congruent triangles will do it (Part I p.108, item 9)
• Decide whether "opposite sides parallel and equal" is a workable definition of a rectangle (Part I p.94, item 5)

Examples worth working on the board

Values marked the chapter's own are worked out in the running text of §4.3, not in an exercise, and may be shown as such.

• Making one that is not a rectangle (Part I, §4.3, p.95). Draw two pairs of parallel lines that cross at angles other than right angles. The chapter's figure shows four lines crossing, with the quadrilateral they enclose labelled A upper left, B upper right, C lower right, D lower left.
• Where the rectangle sits (Part I, §4.3, p.96). A rectangle has both pairs of opposite sides parallel, so it meets the definition; it is a parallelogram whose angles happen to be 90°. The chapter's Venn diagram on that page nests three regions: Square inside Rectangle inside Parallelogram.
• The construction (Part I, §4.3, p.96). Two steps. Step 1: draw AB = 4 cm and AD = 5 cm with 30° between them. Step 2: through D draw a line parallel to AB, through B draw a line parallel to AD, and call the crossing point C. The finished figure has D upper left, C upper right, A lower left, B lower right.
• Deduction 6, the inputs (Part I, §4.3, p.97). AB is parallel to CD, and AD crosses both, so the angles at A and at D total 180°. The chapter's own: with ∠A = 30°, ∠D = 150°. Then AD is parallel to BC with AB and CD crossing them, giving the other two pairs, and the chapter's own values ∠B = 150° and ∠C = 30°. All four adjacent pairs total 180°, and the two opposite pairs are equal.
• Deduction 6 in general (Part I, §4.3, p.97). The chapter reruns the same argument on a parallelogram lettered with R upper left, A upper right, P lower left, E lower right, taking the angle at P to be x. The chapter's own: ∠R comes to 180 − x, then ∠A comes back to x. This is the step that makes the result general rather than a fact about one drawing.
• Deduction 7, the inputs (Part I, §4.3, p.98). Parallelogram ABCD with D upper left, C upper right, A lower left, B lower right, and the diagonal BD drawn. Two triangles sit either side of it and share it. Their other data: one pair of angles equal because they are the parallelogram's opposite angles; a second pair equal because AD is parallel to BC and BD crosses them, so those are alternate angles. AAS applies, and the opposite sides come out equal.
• The completed parallelogram (Part I, §4.3, p.98). The chapter's filled-in figure carries 4 cm on the top and bottom, 5 cm on the two slanted sides, 30° at A and at C, and 150° at B and at D.
• Deduction 8, the inputs (Part I, §4.3, pp.98–99). A parallelogram lettered EASY — A upper left, S upper right, E lower left, Y lower right — with both diagonals drawn and crossing at O. Compare the triangle on AE with the triangle on YS. The supplied facts: AE and YS are equal as opposite sides; two pairs of angles are equal as alternate angles across the parallel sides. ASA applies, so O halves both diagonals.
• The negative result (Part I, §4.3, p.98). The chapter asks the student to check the constructed parallelogram and reports that the diagonals need not come out equal.
• The two vertex-order questions. Part I p.98: is it wrong to write ∆ABD ≅ ∆CBD? Part I p.99: is it wrong to write ∆AOE ≅ ∆SOY? Both are inputs.
• The open question (Part I, §4.3, p.99, last line). Do the diagonals meet at any particular angle? The chapter leaves it and picks it up in §4.4.

Figures to have open

• The four crossing lines of Part I p.95 with the enclosed quadrilateral labelled. Standard schematic.
• The nested Venn diagram of Part I p.96: Square inside Rectangle inside Parallelogram, three labels.
• The two construction steps of Part I p.96 as a step-by-step build, with 4 cm, 5 cm and 30° visible throughout.
• Parallelogram ABCD with the diagonal BD, the two triangles distinguishable, and the single-arc and double-arc angle markings the chapter uses on Part I p.98.
• The completed parallelogram of Part I p.98 with all eight labels.
• Parallelogram EASY with both diagonals and the crossing point O, marked as on Part I p.99.
• A deliberately long, thin parallelogram for the diagonal-inequality demonstration. This one is added here; the chapter asks the student to check their own drawing instead.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 4, "Quadrilaterals", §4.3 "More Quadrilaterals with Parallel Opposite Sides", Part I pp.95–99. The section starts in the upper half of Part I p.95 and ends at the foot of Part I p.99, carrying Deduction 6 (Part I p.97), Deduction 7 (Part I pp.97–98) and Deduction 8 (Part I pp.98–99), plus the four numbered parallelogram properties on Part I p.98.
• Venn diagram placing the rectangle inside the parallelogram: Part I p.96.
• The parallelogram properties restated in the chapter summary: Part I p.109.
• Exercises: Part I p.102, items 1(i), 1(ii) and 2; Part I p.108, item 9; Part I p.109, items 11(iii) and 11(v); Part I p.94, item 5.