⬠Polygons & Quadrilaterals

Surveyor Kit prices a roof by its corners, and she only ever needs one fact to do it: a polygon with n sides splits into (n - 2) triangles.

Pick any corner of a pentagon and draw the diagonals from it. Two of the sides already reach that corner, so the diagonals reach the other corners and cut the pentagon into three triangles. A hexagon gives four, an octagon six. The triangles use up every scrap of the inside and nothing is counted twice, so:

interior angle sum = (n - 2) times 180°

A pentagon: (5 - 2) times 180 = 540°. An octagon: (8 - 2) times 180 = 1080°.

There is a second angle at every corner. Extend one side past the corner and the angle between that extension and the next side is the exterior angle. It sits on a straight line with the interior angle, so at every single corner:

interior angle + exterior angle = 180°

And here is the surprise that never stops being useful: the exterior angles of any polygon add to 360°, whatever the number of sides. Walk once round the outside and at each corner you turn by the exterior angle; by the time you are back where you started, facing the way you set off, you have turned through exactly one full turn.

A regular polygon has all sides equal and all angles equal, so both totals get shared out evenly:

each exterior angle = 360 / n each interior angle = (n - 2) times 180 / n

Run it backwards and you can find n from a single angle. Given an interior angle of 150°, the exterior angle is 180 - 150 = 30°, and 360 / 30 = 12, so it is a 12-gon. Always turn an interior angle into its exterior angle first — the 360° belongs to the exterior angles, never to the interior ones.

A parallelogram is defined by one thing only: both pairs of opposite sides are parallel. Everything else is a consequence, and it is worth seeing where each one comes from rather than memorising a list.

Label the corners A, B, C, D in order round the shape, so AB is opposite DC and AD is opposite BC.

• ◆Consecutive angles are supplementary. AD and BC are parallel, and the side AB crosses both of them. Angle A and angle B are co-interior angles between parallel lines, so angle A + angle B = 180°. The same argument works at every side, so *any two angles at the ends of one side add to 180°*.
• ◆Opposite angles are congruent. Angle A + angle B = 180° and angle B + angle C = 180°, so angle A = angle C. Nothing more is needed.
• ◆Opposite sides are congruent. Draw the diagonal AC. The alternate angles it makes give two triangles with equal angles and a shared side, so the triangles are congruent, and matching sides give AB = DC and AD = BC.
• ◆The diagonals bisect each other. The same congruent triangles show that the crossing point E is the midpoint of both diagonals: AE = EC and BE = ED.

Those last two are the ones students mix up under pressure. Say them out loud in the right shape: *opposite angles equal, consecutive angles supplementary.* If a question gives you angle A and asks for angle C, copy it. If it asks for angle B or angle D, subtract from 180.

When two angles are given as expressions — angle A = (3x + 15)° and angle C = (2x + 40)° — the property tells you which equation to write. Opposite angles: set the expressions equal. Consecutive angles: make them add to 180. Solve, and then read the question again, because it may want an angle rather than x.

All three are parallelograms, so all three inherit everything above. What separates them is what happens to the diagonals.

• ◆A rectangle is a parallelogram with four right angles. Its diagonals are congruent as well as bisecting each other. So if AC = 24 cm then BD = 24 cm, and each of the four half-diagonals AE, EC, BE, ED is 12 cm.
• ◆A rhombus is a parallelogram with four congruent sides. Its diagonals are perpendicular, and each one bisects the two corner angles it reaches. So if angle ABC = 74°, the diagonal BD splits it into two angles of 37°.
• ◆A square is both at once: congruent diagonals, perpendicular diagonals, angles bisected. A diagonal of a square makes 45° with each side, and the diagonals meet at 90°.

The perpendicular diagonals of a rhombus are a gift, because they build four right triangles whose legs are *half* of each diagonal. If the diagonals are 16 cm and 30 cm, the legs are 8 cm and 15 cm, and each side of the rhombus is the hypotenuse: sqrt(8^2 + 15²) = sqrt(289) = 17 cm. Using 16 and 30 as the legs is the classic slip, and it gives an answer exactly twice too big.

A congruent-diagonals question with expressions works like the parallelogram one: AC = (3x + 7) cm and BD = (5x - 5) cm in a rectangle means 3x + 7 = 5x - 5, so x = 6 and the diagonal is 25 cm. Watch what is asked for — the whole diagonal, half of it, or x itself.

A trapezoid has exactly one pair of parallel sides, called the bases; the other two are the legs. Because the bases are parallel and each leg crosses them both, the two angles at the ends of a leg are supplementary. In an isosceles trapezoid the legs are congruent, and then the two angles sitting on the same base are congruent and the diagonals are congruent too.

The midsegment joins the midpoints of the two legs, runs parallel to both bases, and is their average:

midsegment = (base + base) / 2

Bases of 18 cm and 24 cm give a midsegment of (18 + 24) / 2 = 21 cm — always between the two bases, which is a quick check that you halved. Running it backwards, a midsegment of 20 cm with one base of 14 cm means the bases add to 40 cm, so the other base is 26 cm. Double the midsegment *first*, then subtract.

A kite has two pairs of congruent *adjacent* sides: AB = AD and CB = CD. The diagonal AC joining those two corners is its line of symmetry, which forces three things:

• ◆angle B and angle D — the pair between the sides of different lengths — are congruent;
• ◆the diagonals are perpendicular;
• ◆AC bisects BD, so BE = ED. (BD does *not* bisect AC, which is what makes a kite feel lopsided.)

So with angle A = 70° and angle C = 100°, the two equal angles share what is left of 360°: (360 - 170) / 2 = 95° each. And because the diagonals are perpendicular with one of them bisected, a side comes straight from the Pythagorean theorem: BD = 16 cm and AE = 15 cm give BE = 8 cm and AB = sqrt(15^2 + 8²) = 17 cm.

Read downwards and each name keeps everything above it and adds one demand: quadrilateral, then parallelogram (both pairs of opposite sides parallel), then rectangle (add right angles) or rhombus (add equal sides), then square (both). Trapezoids and kites hang off the side, because neither is a parallelogram.

That tree is why a classification question asks for the name that is most precise. "Both pairs of opposite sides parallel and the diagonals congruent" is true of a parallelogram, but rectangle uses every word of it, so rectangle is the answer. The other classification question runs the other way — *which statement is NOT always true* — and the way to break a statement is to sketch an extreme example. A long, stretched rhombus shows at once that a rhombus need not have congruent diagonals.

How the questions here want their answers:

• ◆Angles are whole numbers of degrees. Type 108, not 108° and not 108 degrees.
• ◆Lengths are whole numbers of centimetres. Type 17.
• ◆A number of sides is a whole number. Type 12.
• ◆x is a whole number. When a question shows expressions and asks "what is the value of x", give x. When it asks for an angle or a length, solve for x first and then substitute — handing in x where a measure was asked for is the single most common way to lose a correct piece of work.
• ◆Classification questions are multiple choice: pick the option, and remember the most precise name wins.

Corners are always named in order around the figure, so in ABCD the side AB is opposite DC, angle A is opposite angle C, and E is the point where the diagonals cross. Surveyor Kit will hand you the measurements in that order too.