Which quadrilaterals you can build by joining two triangles

What to assume they know

• The parallelogram: everything that follows from "opposite sides parallel" — the parallelogram, and equal opposite sides
• The rhombus, and what its diagonals do — the rhombus, and the four-equal-sides condition
• What the diagonals alone tell you about a quadrilateral — the joined edge read as a diagonal
• Equilateral, isosceles and scalene triangles, and their angle patterns
• That two congruent triangles have equal matching sides and angles
• That the angles of a triangle total 180°, and the angles of a quadrilateral 360°

What they should be able to do

• Join two congruent cardboard triangles along a matching edge and name the quadrilateral produced, with a justification
• Explain why the joined edge is a diagonal of the new figure and not a side
• Explain why turning the second triangle round gives equal opposite sides, and turning it over gives equal neighbouring sides
• Predict which quadrilateral appears from a given triangle and a given joining edge, before cutting anything out
• Enumerate the different quadrilaterals obtainable from one scalene triangle, and say why an equilateral triangle offers so much less variety
• Give all four sides and all four angles of the quadrilateral made from two equilateral triangles of stated side length
• Recognise that a figure formed this way can cave inwards, and say when

Where it usually goes wrong

• "You can join them along any two sides." Only edges of equal length can be pressed together without leaving a gap or an overhang, which is why a scalene triangle offers three choices and not nine.
• "The joined edge is a side of the new shape." It is inside the figure. Students who count it as a side get a five-sided figure and lose the plot.
• "Two triangles always make a parallelogram." They do when one is turned round. Turn it over instead and the equal sides land next to each other rather than opposite, giving a kite.
• "Two equilateral triangles make a square." A common answer, and wrong: the angles are built from 60° pieces, so the corners come out 60° and 120°.
• "The shape must be convex." Not always. Turning a triangle over across an edge next to an obtuse angle doubles that angle past a straight angle and the figure caves inwards. The chapter draws such a quadrilateral twice, on Part I p.82 and Part I p.108.
• "Naming it is the answer." Every question on these two pages asks for a justification as well. The name comes from checking a property — four equal sides, or equal opposite sides — not from recognising a silhouette.

Questions to check understanding

• Name the quadrilateral formed by joining two given congruent triangles, with a justification that names the property used
• Give every side and every angle of the figure made from two equilateral triangles of a stated side (Part I p.107, item 1)
• List the different quadrilaterals obtainable from one scalene triangle, and say which joining produced each
• Work backwards: given a quadrilateral, say which triangle it is built from and along which edge
• Explain why a particular pair of triangles cannot be joined along a stated pair of edges
• Decide whether the 360° total still holds for a figure that caves inwards (Part I p.108, item 10)

Examples worth working on the board

The chapter poses all of these as open questions and prints no answers; the analysis flagged as added here is a map as a check, not a script.

• Two equilateral triangles, 8 cm (Part I, §4.5, p.104, item 1). Two identical cutouts, every side 8 cm. The chapter shows them separate, asks whether they can be joined into a quadrilateral, then shows the result: a four-sided figure carrying 8 cm on all four sides with an 8 cm segment running through the middle from one corner to the opposite one. The question printed underneath asks for the type and a justification.
• Two isosceles triangles, 8 cm, 8 cm, 6 cm (Part I, §4.5, p.104, item 2). The chapter asks for the different ways they can be joined, then shows two of them. The first result carries 8 cm on all four sides with a 6 cm segment through the middle. The second carries 6 cm at the top and bottom, 8 cm on both slanted sides, and an 8 cm segment through the middle. Both are followed by "what quadrilaterals are these?" and a demand for justification.
• Two scalene triangles, 6 cm, 9 cm, 12 cm (Part I, §4.5, p.105, item 3). The chapter shows the pair and asks the same two questions — how many different ways, and which quadrilaterals result — without showing any of the results on that page. One of them appears at the top of §4.6.
• The kite that opens §4.6 (Part I, p.105). Item 3 itself shows none of the results; one appears further down this same page, under the §4.6 heading. Joining the 6-9-12 pair along their 12 cm edges gives a figure carrying 6 cm on the two upper sides, 9 cm on the two lower sides, and 12 cm running down the middle.
• The 4 cm version (Part I, §4.6 exercise, p.107, item 1). Two equilateral triangles of side 4 cm; the exercise asks for every side and every angle of the quadrilateral obtained.
• An added analysis, for a reviewer to check and as a check to plan against. Each cutout can be laid against its twin in two ways along any one matching edge: turned round through half a turn about the middle of that edge, or turned over across it. * Turned round — the two sides that were not joined appear once on each side of the figure, so opposite sides match and a parallelogram results. Its two side lengths are the two unjoined lengths, and the joined edge is a diagonal. * Turned over — each unjoined side lands next to its own copy, so two neighbouring pairs match and a kite results. Its side lengths are the two unjoined lengths, each used twice, and the joined edge is the diagonal that splits it symmetrically. For the 6-9-12 triangle that gives at most six figures, three parallelograms with side pairs 6 and 9, 6 and 12, 9 and 12, and three kites. Only the kite built on the 12 cm edge comes out without caving inwards — the triangle's largest angle faces its 12 cm side and is more than 90°, so turning over across either shorter edge doubles that angle past a straight angle. For the 8-8-6 triangle the count drops to three distinct figures, because turning round and turning over across the 6 cm edge give the same rhombus. For the equilateral triangle it drops to one.

Figures to have open

• The two equilateral cutouts of Part I p.104 with 8 cm on every side, and the joined result with four 8 cm sides and the 8 cm interior segment.
• The two isosceles cutouts, 8 cm, 8 cm and 6 cm, and both joined results exactly as Part I p.104 labels them — the first with four 8 cm sides and a 6 cm interior segment, the second with 6 cm top and bottom, 8 cm slants and an 8 cm interior segment.
• The two scalene cutouts, 6 cm, 9 cm and 12 cm (Part I p.105), and the kite built from them with 6, 6, 9, 9 and the 12 cm interior segment.
• A movement of one cutout being turned round and, separately, turned over, with the two results shown together. This is an added figure and it carries section 7.
• A grid of the possibilities for the scalene case, with the caved-in ones marked as such. Also not in the book.
• No photograph is needed; all of these are line figures.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 4, "Quadrilaterals", §4.5 "Playing with Quadrilaterals", unnumbered bold subheading "Joining Triangles", Part I pp.104–105, items 1, 2 and 3.
• The kite built from the 6-9-12 pair, at the head of §4.6 "Kite and Trapezium", Part I p.105.
• Exercise: Part I p.107, item 1.
• The caved-in quadrilateral used for the angle-total question: Part I p.108, item 10; a second one appears on the chapter's opening page, Part I p.82, figure (iii).