Quadrilaterals

A quadrilateral is a closed figure made from four straight line segments. That is the entire definition: four sides joined end to end, with no gaps, no curves and no crossing over. The name itself is the memory hook, because quadri means four and lateral means side.

Every quadrilateral carries four of almost everything. It has four sides, four vertices (the corner points) and four interior angles. It also has exactly two diagonals, which are the segments joining opposite corners. In quadrilateral ABCD the sides are AB, BC, CD and DA, and the two diagonals are AC and BD. There is no third diagonal, because every other pair of vertices you could join is already a side.

Naming matters far more than most students expect. You must read the letters in order as you travel around the boundary, either clockwise or anticlockwise. So ABCD is a correct name for the figure, while ACBD is not, because A and C are opposite corners rather than neighbours. If the name is written out of order, every later step about opposite sides or opposite angles is built on the wrong pairs.

Two words come up in almost every question:

• Adjacent sides share a vertex. In ABCD, sides AB and BC are adjacent because they meet at B.
• Opposite sides never touch. AB and CD are opposite, and so are BC and DA.

The same idea works for angles. In ABCD, angle A and angle C form a pair of opposite angles, while angle A and angle B are adjacent angles. Whenever a property speaks about opposite angles being equal, it means pairs like A and C, never A and B.

Quadrilaterals also split into two families by shape. In a convex quadrilateral every interior angle is smaller than 180 degrees, and both diagonals lie completely inside the figure. In a concave quadrilateral one interior angle is reflex, which means bigger than 180 degrees, the outline has a dent in it, and one diagonal falls outside the shape. Nearly every question at this level uses convex quadrilaterals, but the angle sum rule in the next section holds for both kinds.

Here is a plain use of all this. A farmer fences a plot shaped like quadrilateral PQRS, with PQ = 18 m, QR = 25 m, RS = 20 m and SP = 15 m. The perimeter is just the total of the four sides: 18 + 25 + 20 + 15 = 78 m. If fencing wire costs ₹45 per metre, the bill is 78 × 45 = ₹3,510. Notice that no special property was needed anywhere. Perimeter is always the total of the four side lengths, whatever kind of quadrilateral you are looking at.

You already know that the three angles of a triangle add up to 180 degrees. That single fact is enough to prove the most useful rule in this whole chapter: the four interior angles of any quadrilateral add up to 360 degrees.

Here is the proof, and it is short. Take quadrilateral ABCD and draw a diagonal that runs inside the figure, say AC. In a convex quadrilateral either diagonal will do; in a concave one you must use the diagonal drawn from the reflex corner, since the other one falls outside. That one line cuts the quadrilateral into two triangles, ABC and ACD. In triangle ABC the angles add to 180 degrees, so angle BAC + angle B + angle BCA = 180. In triangle ACD the angles also add to 180 degrees, so angle CAD + angle D + angle ACD = 180. Now add the two statements. On the left, angle BAC and angle CAD join up to make the full corner angle A, and angle BCA and angle ACD join up to make the full corner angle C. What is left is exactly angle A + angle B + angle C + angle D = 180 + 180 = 360.

The two triangles cover every part of the quadrilateral once and only once, and no angle gets counted twice, which is why the two 180s can simply be added. If you prefer a hands-on version, cut any four-sided shape out of paper, tear off the four corners and place them side by side around a single point. They fit exactly around the point with no overlap and no gap, and one full turn about a point is 360 degrees.

Once you trust the rule, most angle questions become arithmetic.

Example 1. Three angles of a quadrilateral are 85 degrees, 100 degrees and 75 degrees. Find the fourth. Add the three known angles: 85 + 100 + 75 = 260. The fourth angle is 360 - 260 = 100 degrees. Check by adding all four: 85 + 100 + 75 + 100 = 360, so the answer stands.

Example 2. The four angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. Find them. A ratio is a set of shares, so write the angles as 1x, 2x, 3x and 4x. Then x + 2x + 3x + 4x = 360, which gives 10x = 360 and x = 36. The four angles are 36 degrees, 72 degrees, 108 degrees and 144 degrees. Check the total: 36 + 72 = 108, then 108 + 108 = 216, then 216 + 144 = 360. Correct.

Example 3. Three angles of a quadrilateral are equal and the fourth is 120 degrees. Find the equal angles. Together the three equal angles must make 360 - 120 = 240 degrees, so each one is 240 ÷ 3 = 80 degrees. The four angles are 80, 80, 80 and 120 degrees, which do add to 360.

One more fact is worth carrying, and this one is for convex quadrilaterals. If you extend each side in turn and mark one exterior angle at each vertex, those four exterior angles also add up to 360 degrees. The reason is that each exterior angle is 180 degrees minus its own interior angle, so all four together come to 4 × 180 - 360 = 720 - 360 = 360 degrees.

Special quadrilaterals are not separate inventions. Each one is an ordinary quadrilateral with an extra condition stapled on, and the conditions stack. Read the list from the loosest condition to the tightest and the whole family tree falls into place.

• Trapezium — a quadrilateral with a pair of parallel sides. The parallel sides are called the bases and the other two are the legs. If the two legs are equal in length and are not themselves parallel, it is an isosceles trapezium, and then the two angles on each base are equal and the diagonals come out equal too.
• Parallelogram — both pairs of opposite sides are parallel. That single condition forces a lot: opposite sides also become equal, opposite angles become equal, and the diagonals cut each other in half.
• Rhombus — a parallelogram in which all four sides are equal. Its angles need not be right angles, so a rhombus usually looks like a pushed-over square, which is why a diamond on a playing card is a rhombus.
• Rectangle — a parallelogram with one right angle. One is enough, because in a parallelogram the adjacent angle is 180 minus that, which is also 90, and the opposite angles copy them. So one right angle drags the other three along.
• Square — a quadrilateral that is a rhombus and a rectangle at the same time: four equal sides and four right angles. It therefore inherits every property of both.
• Kite — two pairs of adjacent sides are equal. In kite ABCD that means AB = AD and CB = CD. Note the word adjacent: in a parallelogram the equal sides are opposite, and that is the whole difference between the two shapes.

The family relationships run one way only, and this is where answers most often go wrong. Every square is a rectangle, but not every rectangle is a square: a 10 cm by 4 cm rectangle has four right angles and unequal sides. Every square is a rhombus, but not every rhombus is a square: a rhombus with angles of 60 and 120 degrees has four equal sides and no right angle. Every rectangle and every rhombus is a parallelogram, but a parallelogram need not be either of them. When you have to judge such a statement, do not argue in words; produce a single counter-example and you are done.

Two definitions differ from book to book, so it helps to know both readings. If a trapezium is taken to have at least one pair of parallel sides, then every parallelogram counts as a trapezium as well; if it must have exactly one pair, parallelograms are excluded. In the same way, some books let a rhombus count as a kite, since four equal sides do give two pairs of adjacent equal sides, while others insist the two pairs be of different lengths. If a question turns on this point, write down which reading you are using so your reasoning is clear.

Here is the working list. Learn it as short sentences rather than as a table, because in a question you will be recalling one line at a time.

• Parallelogram. Opposite sides are equal and parallel. Opposite angles are equal, so angle A = angle C and angle B = angle D. Any two adjacent angles add to 180 degrees, because they are co-interior angles between a pair of parallel sides.
• Rhombus. Everything a parallelogram has, plus all four sides equal. So the perimeter is 4 times a side, and the angles still come in equal opposite pairs that are supplementary with their neighbours.
• Rectangle. Everything a parallelogram has, plus all four angles equal to 90 degrees. Opposite sides are equal, but adjacent sides usually are not.
• Square. Four equal sides and four right angles at once.
• Kite. Two pairs of adjacent sides equal. The pair of angles caught between the unequal sides are equal: in kite ABCD with AB = AD and CB = CD, angle B = angle D. Angles A and C are usually different from each other.
• Trapezium. With AB parallel to DC, each leg acts as a transversal across the parallel pair, so angle A + angle D = 180 degrees and angle B + angle C = 180 degrees. In an isosceles trapezium the two angles on the same base are equal.

Example 1. In parallelogram ABCD, angle A = (3x + 10) degrees and angle B = (2x - 5) degrees. Find both angles. A and B are adjacent, so they add to 180: (3x + 10) + (2x - 5) = 180, giving 5x + 5 = 180, then 5x = 175 and x = 35. So angle A = 3(35) + 10 = 105 + 10 = 115 degrees and angle B = 2(35) - 5 = 70 - 5 = 65 degrees. Check: 115 + 65 = 180. The full set of angles is 115, 65, 115 and 65 degrees, adding to 360.

Example 2. In kite ABCD with AB = AD and CB = CD, angle A = 120 degrees and angle C = 50 degrees. Find angles B and D. The four angles total 360, so angle B + angle D = 360 - 120 - 50 = 190 degrees. Since the kite forces angle B = angle D, each one is 190 ÷ 2 = 95 degrees. Check: 120 + 95 + 50 + 95 = 360.

Example 3. In trapezium ABCD with AB parallel to DC, angle A = 65 degrees and angle B = 100 degrees. Find angles C and D. Along leg AD, angle A + angle D = 180, so angle D = 115 degrees. Along leg BC, angle B + angle C = 180, so angle C = 80 degrees. Check the total: 65 + 100 + 80 + 115 = 360.

Example 4. The perimeter of a rhombus is 52 cm. Find its side. All four sides are equal, so each side is 52 ÷ 4 = 13 cm. It is a one-step calculation, but it often sits buried inside a longer problem, so learn to spot it quickly.

If someone hides a shape and tells you only about its diagonals, you can usually name it. Four separate powers are on offer, and each special quadrilateral owns a different set of them. Call the point where the diagonals cross O.

• Parallelogram. The diagonals bisect each other, meaning OA = OC and OB = OD. They are usually neither equal nor perpendicular.
• Rectangle. The diagonals bisect each other and are equal, so AC = BD. Because both are cut in half at O, all four pieces OA, OB, OC and OD are the same length, which makes O the same distance from every corner.
• Rhombus. The diagonals bisect each other and meet at right angles. They also bisect the angles of the rhombus that they pass through. They are not equal to each other unless the rhombus happens to be a square.
• Square. All of it: equal, bisecting each other, perpendicular, and bisecting the corner angles into 45 degree halves.
• Kite. Only one diagonal is special. In kite ABCD with AB = AD and CB = CD, the diagonal AC is the perpendicular bisector of BD, so BO = OD and the crossing is at 90 degrees, but AO and OC are usually different lengths.
• Trapezium. A plain trapezium gives you nothing. An isosceles trapezium gives equal diagonals.

Because a rhombus, a square and a kite all produce right angles at O, they hand you right-angled triangles for free, and that is where the Pythagoras relation earns its keep.

Example 1. In parallelogram ABCD the diagonals meet at O, with OA = (3x - 1) cm and OC = (x + 5) cm. Find AC. Diagonals bisect each other, so OA = OC: 3x - 1 = x + 5, giving 2x = 6 and x = 3. Then OA = 3(3) - 1 = 8 cm and OC = 3 + 5 = 8 cm, which agree, so AC = 8 + 8 = 16 cm.

Example 2. A rhombus has diagonals of 16 cm and 30 cm. Find its side and perimeter. The diagonals bisect each other at right angles, so the half-diagonals are 8 cm and 15 cm and they form a right-angled triangle with a side of the rhombus as hypotenuse. Side = square root of (8² + 15²) = square root of (64 + 225) = square root of 289 = 17 cm. Perimeter = 4 × 17 = 68 cm.

Example 3. In rectangle ABCD the diagonals meet at O and angle OAB = 35 degrees. Find angle AOB and angle BOC. Since the diagonals are equal and bisect each other, OA = OB, so triangle OAB is isosceles and angle OBA = 35 degrees too. Then angle AOB = 180 - 35 - 35 = 110 degrees. Angles AOB and BOC sit on the straight line AC, so angle BOC = 180 - 110 = 70 degrees.

Example 4. In kite ABCD, AB = AD = 17 cm, CB = CD = 10 cm and BD = 16 cm. Find AC. Diagonal AC bisects BD at right angles, so BO = OD = 8 cm. In right-angled triangle ABO, AO = square root of (17² - 8²) = square root of (289 - 64) = square root of 225 = 15 cm. In right-angled triangle CBO, CO = square root of (10² - 8²) = square root of (100 - 64) = square root of 36 = 6 cm. So AC = AO + OC = 15 + 6 = 21 cm.

Example 5. In rhombus ABCD, angle A = 116 degrees. Find angle BAC and angle ABC. Diagonal AC bisects angle A, so angle BAC = 116 ÷ 2 = 58 degrees. Angle B is adjacent to angle A in a parallelogram, so angle ABC = 180 - 116 = 64 degrees. As a check, triangle ABC has AB = BC, so angle BCA must also be 58 degrees, and 58 + 58 + 64 = 180.

Almost every question in this chapter is solved the same way, so it is worth turning it into a habit:

1. Name the shape and write down which special quadrilateral it is.
2. List the quantities you are given and mark them on a rough figure.
3. Choose the one property that connects what you know to what you want.
4. Form an equation and solve it.
5. Check: angles must add to 360, adjacent angles of a parallelogram must add to 180, and lengths must be positive and sensible.

Example 1 (all four angles unknown). The angles of a quadrilateral are x, (x + 15), (x + 25) and (x + 40) degrees. Find them. Their sum is 360, so x + (x + 15) + (x + 25) + (x + 40) = 360. Collect the x terms to get 4x, and the numbers to get 15 + 25 + 40 = 80, so 4x + 80 = 360. Then 4x = 280 and x = 70. The angles are 70, 85, 95 and 110 degrees, and 70 + 85 + 95 + 110 = 360.

Example 2 (parallelogram, angles in a ratio). Two adjacent angles of a parallelogram PQRS are in the ratio 3 : 2. Find all four angles. Write them as 3x and 2x. Adjacent angles of a parallelogram are supplementary, so 3x + 2x = 180, giving 5x = 180 and x = 36. So angle P = 3(36) = 108 degrees and angle Q = 2(36) = 72 degrees. Opposite angles are equal, so angle R = 108 degrees and angle S = 72 degrees. The total is 108 + 72 + 108 + 72 = 360.

Notice the trap in that example. Because the two angles were adjacent, the ratio parts were shared out of 180, not 360. If a question gives all four angles in a ratio, the parts come out of 360 instead. Read carefully before you divide.

Example 3 (rectangle, finding a diagonal). A rectangle is 24 cm long and 7 cm wide. Find the length of a diagonal. The diagonal is the hypotenuse of a right-angled triangle whose legs are the length and the breadth, so diagonal = square root of (24² + 7²) = square root of (576 + 49) = square root of 625 = 25 cm.

Example 4 (parallelogram, finding a side). The perimeter of a parallelogram is 48 cm and one side is 14 cm. Find the adjacent side. Opposite sides are equal, so the perimeter is 2 × (sum of two adjacent sides): 2(14 + b) = 48, so 14 + b = 24 and b = 10 cm. The four sides are 14, 10, 14 and 10 cm, and 14 + 10 + 14 + 10 = 48.

Example 5 (exterior angles). Three exterior angles of a convex quadrilateral are 70, 80 and 120 degrees. Find the fourth, and then the interior angles. The four exterior angles total 360, so the fourth is 360 - 70 - 80 - 120 = 90 degrees. Each interior angle is 180 minus its exterior angle, giving 110, 100, 60 and 90 degrees, and those four do add to 360.

Two habits will save you again and again. First, always sketch the figure, even roughly, because an unmarked question is where sides and angles get mixed up. Second, always run the final check; adding four angles takes five seconds and catches almost every slip.

The mathematics here is not hard, but the same handful of slips keeps turning a correct method into a wrong answer. Read these once before you start practising.

Slip 1: using 180 instead of 360. The triangle rule is so familiar that it leaks across. A quadrilateral has four angles and they total 360 degrees. If your four angles add to 180, you have used the wrong rule.

Slip 2: swapping opposite and adjacent. In a parallelogram, opposite angles are equal and adjacent angles are supplementary. Mixing them up leaves you writing 65 degrees where the answer should have been 180 - 65 = 115 degrees. Before applying either rule, check on your figure whether the two angles touch at a vertex or sit across the shape.

Slip 3: assuming a rhombus has equal diagonals. A rhombus guarantees you perpendicular diagonals, not equal ones. A rectangle guarantees you equal diagonals, not perpendicular ones. Only the square has both, because it is a rhombus and a rectangle at once. Keep the two words separate in your head: equal belongs to the rectangle branch, perpendicular belongs to the rhombus branch.

Slip 4: bisecting both diagonals of a kite. In a kite only one diagonal is cut in half, and it is easy to pick the wrong one. In kite ABCD with AB = AD and CB = CD, the diagonal AC joins the two vertices where the equal sides meet, and AC is the cutter: it is BD that gets bisected, so BO = OD while AO and OC are usually different. Marking both diagonals as bisected turns a kite question into a rhombus answer.

Slip 5: dividing the wrong total in ratio questions. If the ratio describes all four angles, share out 360. If it describes two adjacent angles of a parallelogram, share out 180. If it describes two opposite angles of a parallelogram, the ratio has to be 1 : 1, because opposite angles of a parallelogram are equal. In a quadrilateral with no special property, though, opposite angles need not be equal at all, so there a ratio like 2 : 3 is perfectly possible.

Slip 6: careless naming. Writing the vertices out of order makes AB and CD look adjacent when they are opposite. Always go round the boundary in one direction.

Now the checks. They take seconds and they catch nearly everything:

• Add your four angles. If the total is not exactly 360 degrees, something is wrong, and you can usually spot which angle by testing the adjacent pairs.
• For a parallelogram, rhombus, rectangle or square, test that every pair of adjacent angles adds to 180.
• For a rhombus, check that 4 times your side equals the perimeter you were given.
• For any length found with the Pythagoras relation, check the hypotenuse is the longest of the three. If a half-diagonal comes out longer than the side of the rhombus, you have subtracted where you should have added.

One last piece of advice about the family tree. When a true-or-false question asks whether every shape of one kind is also of another kind, do not try to argue it out in sentences. Draw or describe one counter-example. A rectangle measuring 6 cm by 4 cm settles any claim that all parallelograms are rhombuses, and it settles it faster than a paragraph of reasoning ever will.