The rhombus, and what its diagonals do

What to assume they know

• The parallelogram: everything that follows from "opposite sides parallel" — the parallelogram and its four properties, all of which the rhombus will inherit
• Properties of a rectangle, and why the square is the special case — the square, and the fact that its diagonals cross at right angles
• That a triangle with two equal sides has two equal angles facing them
• Alternate angles: if they are equal, the two lines are parallel
• The congruence conditions, especially SSS
• Using a compass to carry a length and to cut arcs from two centres

What they should be able to do

• Construct a quadrilateral with four equal sides that is not a square, starting from two equal sides at a chosen angle and cutting arcs of that same length
• State the definition of a rhombus and explain why it does not mention angles
• Prove that a diagonal of a rhombus makes four equal angles with the sides, using the isosceles triangles and one congruence
• Explain why those four equal angles establish that the opposite sides are parallel, and hence that every rhombus is a parallelogram
• Find all four angles of a rhombus from one given angle, by two different routes
• Place the rhombus in a Venn diagram with the parallelogram, the rectangle and the square, and say what the overlap of rectangle and rhombus is
• Prove that the diagonals of a rhombus cross at right angles
• State which rhombus properties are inherited from the parallelogram and which are new

Where it usually goes wrong

• "A rhombus is a tilted square." Only if you also tilt one pair of angles away from 90°, which is the whole difference. A square is one rhombus among many, not the standard one.
• "Equal sides make equal angles." They do not — the constructed rhombus has 50° at two corners and 130° at the other two. Students transfer the triangle result, where equal sides really do force equal angles, into a quadrilateral, where it fails.
• "The rhombus is parallel-sided because it looks it." The chapter proves it, and the proof runs through the four equal angles a diagonal makes. That route is the content of the explanation; asserting the conclusion skips the only interesting part.
• "Diagonals crossing at right angles make a rhombus." Not on their own — a kite does that too without having four equal sides, which is why Part I p.109 sets it as a true-or-false item. The diagonals must also halve each other.
• "The diagonals of a rhombus are equal." They are not, except in the square. Long thin rhombuses make this obvious.
• "Every property in the list needed its own proof." Three of the six are inherited from the parallelogram the moment the rhombus is shown to be one.
• "The overlap of rectangle and rhombus is some new shape." It is the square, and the chapter's Venn diagram says so with an arrow.

Questions to check understanding

• Find the remaining angles of a rhombus given one angle between a side and a diagonal — the form of Part I p.102, items 1(iii) and 1(iv)
• Construct a rhombus from the lengths of its two diagonals (Part I p.102, item 3)
• True or false with justification: a quadrilateral whose diagonals are perpendicular must be a rhombus (Part I p.109, item 11(iv))
• Decide whether four equal sides plus one right angle force a square, by reasoning and by construction (Part I p.108, item 8)
• Name the shape you get when two equilateral triangles are pressed together along a matching side, and justify the name (Part I p.104 and Part I p.107, item 1)
• Say which of a rhombus's properties would survive if the equal-side condition were dropped, and which would not
• Place a rhombus, a rectangle and a square correctly in a Venn diagram

Examples worth working on the board

Values marked the chapter's own are worked in the running exposition of §4.4 and may be shown as the chapter's.

• The construction (Part I, §4.4, p.99). Draw AD and AB equal in length, with 50° between them and A at the lower left. Set a compass to the length of AB, and from B and from D cut arcs; where they meet is C. The chapter states that any angle below 180° would have served in place of 50°, which is what makes the family large.
• Deduction 9, the inputs (Part I, §4.4, p.100). A rhombus lettered GAME — E upper left, M upper right, G lower left, A lower right — with the diagonal AE drawn. Four angles are marked: a and d in the lower triangle, b and c in the upper. Two facts are supplied: in the lower triangle two sides are sides of the rhombus, so a = d; in the upper triangle likewise, so b = c. The chapter then asserts the two triangles are congruent and asks the student for the reason. An added reading, for the student to supply: all three pairs of sides match — two are sides of the rhombus and the third is the shared diagonal — so SSS does it. The conclusion is that all four marked angles are equal and the corners at G and at M are equal.
• The parallel step (Part I, §4.4, p.100). Two small figures, each showing one pair of the four equal angles. Take EM and GA with AE crossing them: the equal pair are alternate angles, so those two sides are parallel. Take GE and AM with AE crossing them: same again. Both pairs of opposite sides are therefore parallel.
• The angles of the constructed rhombus, route one (Part I, §4.4, p.100). Rhombus ABCD with D upper left, C upper right, A lower left, B lower right, 50° at A, and the diagonal DB drawn making four equal angles marked a. Inside the triangle on AD and AB: a + a + 50 = 180. The chapter's own: a = 65°, so the corners at D and B are 130° each and the four angles run 50°, 130°, 50°, 130°.
• Route two (Part I, §4.4, p.101). Having established that a rhombus is a parallelogram, take the parallelogram rules instead: the corners at A and C are equal at 50°, and the corners at D and B come from the adjacent pair totalling 180°. Same four numbers, no isosceles triangle needed. The chapter prints both routes deliberately.
• The Venn diagrams (Part I, §4.4, p.101). Two pictures. First, a region labelled Rectangle overlapping a region labelled Rhombus, with an arrow naming the overlap Square. Second, the same pair drawn inside a larger region labelled Parallelogram, with the overlap again labelled Square.
• Deduction 10, the inputs (Part I, §4.4, p.102). Rhombus GAME with both diagonals drawn and crossing at O. Compare the two triangles either side of the crossing on the segment EO. The chapter asserts they are congruent and asks why. Their two angles at O are then equal and lie on a straight line. Worked: each is 90°.
• The six numbered properties (Part I, §4.4, pp.101–102). All sides equal; opposite sides parallel; adjacent angles totalling 180° and opposite angles equal; diagonals halving each other; diagonals halving the angles; diagonals crossing at 90°.
• Exercise inputs from Part I p.102 (§4.4, "Figure it Out"). Item 1(iii): an equal-sided quadrilateral lettered X upper left, W upper right, V lower right, U lower left, with tick marks on all four sides, the diagonal from X to V drawn, and 30° marked at V between the side VU and that diagonal. Item 1(iv): the same arrangement lettered O upper left, I upper right, E lower right, A lower left, tick marks on all four sides, the diagonal from O to E drawn, and 20° marked at E between the side EA and that diagonal. Item 3: construct a rhombus from diagonals of 4 cm and 5 cm.

Figures to have open

• The two printed stages of Part I p.99 — the two equal sides at 50°, then the closed rhombus — plus a third the explanation adds between them showing the compass arcs from both ends. The arcs must be visible; they are what carries the equal-length condition, and the page does not draw them.
• Rhombus GAME with the diagonal AE and the four marked angles a, b, c, d in the printed positions (Part I p.100).
• The two small parallel-test figures from Part I p.100, each isolating one pair of alternate angles.
• Rhombus ABCD with the diagonal DB, the four equal angles marked, and the finished version carrying 50°, 65°, 65° and 130° in the positions Part I p.100 uses.
• The two Venn diagrams of Part I p.101 as clean schematics with all four labels.
• Rhombus GAME with both diagonals and the crossing point O (Part I p.102).
• The two exercise figures of Part I p.102, items 1(iii) and 1(iv), with tick marks on all four sides, the single drawn diagonal, and the 30° and 20° marks in the printed positions.
• A long thin rhombus for the unequal-diagonal demonstration. Not in the book.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 4, "Quadrilaterals", §4.4 "Quadrilaterals with Equal Sidelengths", Part I pp.99–102. The section opens near the foot of Part I p.99 and runs to the numbered properties and Deduction 10 on Part I p.102, carrying Deduction 9 on Part I p.100.
• The Venn diagrams placing rhombus beside rectangle: Part I p.101.
• The rhombus properties restated in the chapter summary: Part I p.110.
• The whole-chapter Venn diagram, in which the rhombus is the overlap of kite and parallelogram: Part I p.110.
• Exercises: Part I p.102, items 1(iii), 1(iv) and 3; Part I p.107, item 4(iii); Part I p.108, item 8; Part I p.109, item 11(iv).