Quadrilaterals

Chapter at a glance

Chapter 12 of Ganita Manjari Part II asks one big question: can any quadrilateral tile the plane? To answer it, the chapter first defines a quadrilateral precisely, proves the converse tests for parallelograms, uses parallelograms to prove the Midpoint Theorem, the Centroid Theorem and Varignon’s Theorem, and finally shows that every 4-gon tiles the plane.

• 12.1 What exactly is a quadrilateral? (definition, convex vs non-convex)
• 12.2 Parallelograms: Theorem 1 and its converses (Theorems 2–5)
• 12.3 Applications: Midpoint Theorem, its converse, medians and the centroid, Varignon parallelogram
• 12.4 Tiling the plane using any 4-gon

12.1 What exactly is a quadrilateral?

“A figure with four straight sides” is not precise enough. The book shows tricky four-point figures (NOPE, SILY, DART, CUTS, OPENS, BENT) to show why we need three conditions:

1. No three vertices collinear: otherwise two sides lie on one line and you get a triangle or overlapping sides.
2. All four vertices in one plane: a figure like BENT is a non-planar quadrilateral.
3. Not self-intersecting: a figure like CUTS, where sides cross, is a self-intersecting quadrilateral.

Words to know

• Vertices: A, B, C, D. Sides (edges): AB, BC, CD, DA. Diagonals: AC and BD.
• Adjacent vertices: the two endpoints of a side. Adjacent sides: two sides with a common endpoint.
• A quadrilateral has 4 internal angles; angles at adjacent vertices are adjacent angles.
• ABCD can also be written BCDA, CDAB, DABC, DCBA, ADCB, BADC or CBAD (go round in either direction from any vertex), but not in any other order.

Convex and non-convex

Why care about strange 4-gons? Four-bar linkages in machines can take convex, non-convex and even self-intersecting shapes as they move, so engineers must handle all of them.

12.2 Parallelograms and their converse tests

A parallelogram is a quadrilateral whose opposite sides are parallel. Rhombuses, rectangles and squares are all parallelograms.

How Theorem 1 is proved

• (a) Diagonal AC is a transversal for AB ∥ DC and AD ∥ BC, so ∆ACD ≅ ∆CAB by ASA. Hence AB = DC and AD = BC.
• (b) With BC ∥ AD and transversal AB, ∠A + ∠B = 180° (co-interior angles). Any two adjacent angles add to 180°, so ∠A = ∠C and ∠B = ∠D.
• (c) Let the diagonals meet at E. Using (a), ∆AED ≅ ∆CEB (ASA/AAS), so EA = EC and EB = ED.

The converses: all true!

Strategy used in the book: run the original proof backwards, giving a fresh justification for every step. (This does not always work: “if x = y then x² = y²” cannot be reversed, since x = 3, y = −3.)

Theorem 3 can also be stated as: “If each pair of adjacent angles of ABCD adds up to 180°, then ABCD is a parallelogram.”

12.3 Applications of parallelograms

Real-life uses: combining forces in physics (a later grade), James Watt’s parallel motion linkage in his steam engine, and the pantograph, which makes exact scaled copies of a drawing.

The Midpoint Theorem

Activity: cut a triangle along the segments joining the midpoints of its sides. You get four congruent triangles, each a smaller copy of the original.

Proof idea. Draw line l through C parallel to BA, meeting line PQ at R. In ∆APQ and ∆CRQ: AQ = CQ, ∠AQP = ∠CQR (vertically opposite), ∠APQ = ∠CRQ (alternate angles, AP ∥ RC). So ∆APQ ≅ ∆CRQ by AAS, giving PQ = QR = PR/2 and CR = AP = BP. By Theorem 5, BCRP is a parallelogram, so PQ ∥ BC and PQ = PR/2 = BC/2.

The extra line l is an auxiliary construction, a classic geometry trick you learn with experience. Exercise 15 gives another proof: extend PQ to S so that ∆APQ ≅ ∆CSQ.

Two proofs in the book. (1) Retrace the Midpoint Theorem proof: BCRP is a parallelogram so CR = BP = PA, and ∆APQ ≅ ∆CRQ, so AQ = QC. (2) Clever proof: let M be the midpoint of AC. By the Midpoint Theorem PM ∥ BC. There is a unique line through P parallel to BC, so line PM = line PQ, hence M = Q. A theorem used to prove its own converse!

Coming in Grade 10: drop “P is the midpoint” and assume only PQ ∥ BC. That gives the Basic Proportionality Theorem.

Medians and the centroid

A median joins a vertex to the midpoint of the opposite side. Experiment: the three medians always seem to be concurrent (meet at one point) and that point cuts each median in the ratio 2 : 1.

Proof idea. Take medians CP and BQ meeting at M. Let Y, X be midpoints of CM and BM. The Midpoint Theorem in ∆ABC and ∆MBC gives PQ ∥ BC ∥ XY and PQ = XY = BC/2. So ∆MPQ ≅ ∆MYX (ASA), giving MQ = MX = XB and MP = MY = YC. Hence CM : MP = BM : MQ = 2 : 1. Repeating with medians AR and BQ (meeting at N) gives BN : NQ = 2 : 1. Only one point divides BQ in a given ratio, so M = N: all three medians pass through M.

Varignon’s Theorem

Proof. With P, Q, R, S the midpoints of AB, BC, CD, DA: the Midpoint Theorem in ∆ABC and ∆ADC gives PQ ∥ AC and SR ∥ AC, so PQ ∥ SR. In ∆BCD and ∆BAD it gives QR ∥ BD ∥ PS. Both pairs of opposite sides are parallel, so PQRS is a parallelogram.

Useful extras from Exercise Set 12.3: PR and QS bisect each other; if AC = BD then PR ⊥ QS; PQRS is a square exactly when AC and BD are equal and perpendicular.

12.4 Tiling the plane using any 4-gon

Tiling means covering the plane with copies of a shape with no gaps and no overlaps (mosaics, M.C. Escher’s art, honeycombs).

• Parallelograms: tilt a rectangular grid of hinged sticks and the lines stay parallel, so any parallelogram tiles the plane.
• Key idea: at every vertex of a tiling the angles add up to 360°, and the four angles of a 4-gon also add up to 360°. So fit 4 copies around a point, using each angle once.
• Method 1 (rotation): rotate a copy by 180° about the midpoint of a side; it fits exactly along that side. Repeat at every edge of every copy.
• Method 2 (Varignon grid): the Varignon parallelograms of all the copies form a parallelogram grid. Draw that grid first, place copies on alternate cells, and the gaps automatically become more copies.
• Method 3 (Exercise 19): slide a copy along its diagonal; shifted copies meet only at vertices and the gaps are again congruent 4-gons.

A regular pentagon cannot tile the plane: each angle is 108°, and 360° is not a multiple of 108°. Some irregular pentagons can tile (the latest type was found in 2015).

Beyond the syllabus: aperiodic tilings never repeat. They helped describe quasicrystals (Dan Shechtman, Chemistry Nobel 2011). In 2023 Smith, Myers, Kaplan and Goodman-Strauss found the “hat”, the first single shape that tiles only aperiodically.

Quick formula and fact sheet

Common mistakes to avoid

• Assuming a theorem’s converse is automatically true. Each converse needs its own proof; the parallelogram ones happen to be true.
• Using “one pair of opposite sides equal” alone; you need equal and parallel for the same pair (Theorem 5).
• Writing the centroid ratio the wrong way round: the longer part (2) is next to the vertex.
• Forgetting to state the congruence test (SSS, SAS, ASA, AAS) and the reason for each equal angle (alternate, vertically opposite, co-interior).

Practice questions (from the textbook)

1. True or false: (i) a parallelogram with a right angle is a rectangle; (ii) a rhombus with perpendicular diagonals is a square; (iii) a parallelogram with equal diagonals is a rectangle. Answers: (i) True (ii) False: every rhombus has perpendicular diagonals (iii) True.
2. Diagonal AC of parallelogram ABCD bisects ∠A. Show it also bisects ∠C and that ABCD is a rhombus. (Ex. 12.2 Q2)
3. In ∆ABC, M and N are midpoints of AB and AC and D is any point on BC. Show that MN bisects AD. (Ex. 12.3 Q2)
4. In parallelogram ABCD, P and Q are points on diagonal BD with DP = BQ. Show that APCQ is a parallelogram. (End Ex. 4)
5. The diagonals of parallelogram ABCD meet at O. A line through O meets AB at P and CD at Q. Show O is the midpoint of PQ. (End Ex. 9)
6. Why can a regular pentagon not tile the plane? (Ex. 12.4 Q1) Each angle is 108°; whole-number multiples of 108° never make exactly 360°.

Chapter summary

• A quadrilateral ABCD: four points, no three collinear, every non-vertex point on exactly one of AB, BC, CD, DA.
• Convex means all internal angles < 180°, or equivalently the diagonals intersect.
• A 4-gon is a parallelogram if: opposite sides are equal, or opposite angles are equal, or the diagonals bisect each other, or one pair of opposite sides is equal and parallel.
• Midpoint Theorem and its converse; medians meet at the centroid in a 2 : 1 ratio; the midpoints of any 4-gon form a parallelogram.
• The plane can be tiled with any given 4-gon.