Class 8 Mathematics Chapter 4: Quadrilaterals

A carpenter with two strips

A carpenter has two thin strips of wood. She wants to join them so that a thread passed round their four endpoints makes a rectangle. One strip is 8 centimetres long. How long must the other be, and where should the two cross?

Turning the question into geometry

The strips are the two diagonals of the shape their endpoints make. So the carpenter is really asking three questions. How long is the second diagonal? Where do the diagonals cross? And at what angle?

The second diagonal

Start from a rectangle and look at two triangles sharing a side. Opposite sides are equal, the angles at the two corners are both right angles, and the shared side is common to both. By the side-angle-side condition the two triangles are congruent, so the diagonals are equal. The other strip must also be 8 centimetres. You can check that by drawing rectangles and measuring, but the congruence argument is what makes it certain.

Where the diagonals cross

Now take the two triangles formed on either side of the crossing point. Vertically opposite angles are equal there. A short angle chase shows a second pair of angles is equal, since each is 90 degrees less the same third angle. The angle-angle-side condition then gives congruence. Corresponding parts of congruent triangles are equal, so the crossing point is the midpoint of each diagonal. We say the diagonals bisect each other, and bisecting simply means cutting into two equal parts.

What about the angle between them?

Try drawing two equal diagonals that bisect each other at 60 degrees. The four angles at the crossing come out as 60, 60, 120 and 120. Each of the four triangles is isosceles, because half a diagonal equals half the other diagonal.

Working the corners out

In the triangle with the 60 degree angle, the two base angles are equal and the three add to 180, so each base angle is 60. In the next triangle the angle at the centre is 120, so its base angles are 30 each. A corner of the quadrilateral is one of each, which is 90.

It works for any angle

Call the angle at the crossing x. The four central angles are x, x, 180 less x, and 180 less x. The base angles work out as 90 less half of x in one triangle and half of x in the next. A corner is their sum, which is 90 whatever x happens to be.

The carpenter has her answer

So any two equal strips joined at their midpoints give a rectangle, at whatever angle they cross. This is not a classroom curiosity. European carpenters square up frames this way, and in Mozambique farmers set out the base of a house by it.

A second definition

That gives a different way to say what a rectangle is. Call it a quadrilateral with two equal diagonals, each cutting the other in half. It sounds nothing like the first definition, yet it picks out exactly the same shapes.

And the first can be trimmed

We first said a rectangle has four right angles and equal opposite sides. Try drawing one with four right angles and unequal opposite sides. You cannot, and congruence shows why: cut along a diagonal and the two triangles must be congruent, which forces the opposite sides to match.

So the shortest definition wins

A rectangle is a quadrilateral whose angles are all right angles. The equal opposite sides need not be stated, because they follow. A good definition asks for no more than it must.

Listing what we know

All four angles are right angles. Opposite sides are equal. Opposite sides are parallel, which follows because co-interior angles across a transversal add to 180. And the diagonals are equal and bisect each other.

Measuring is not proving

This is worth pausing on. Draw a few rectangles, measure, and you will see the diagonals bisect each other every time. That gives a conjecture: a statement you are confident of but have not settled. The thousandth rectangle might be different for all the measuring can say.

The special rectangle

Among rectangles, some have all four sides equal. We call those squares. So a square is a quadrilateral with four right angles and four equal sides, and every square is a rectangle while most rectangles are not squares.

Drawing the relation

A Venn diagram shows this neatly: the set of squares sits entirely inside the set of rectangles. Being a square and being a rectangle are not rival claims, any more than being a Malayali rules out being an Indian.

The carpenter again, for a square

Equal diagonals bisecting each other give a rectangle. For a square we need equal sides too. Congruence settles the angle. The two triangles on one side of a diagonal are congruent by side-side-side. So the two angles at the centre are equal, and together they make a straight angle. Each is therefore 90.

Two more square properties

So the diagonals of a square are equal and bisect each other at right angles. They also bisect the corners. In the triangle cut off by a diagonal two sides are equal, so the two base angles are equal and each is 45 degrees.

Three right angles and a fourth

Can a quadrilateral have three right angles and a fourth angle of some other size? Try it and you will fail, and there is a reason. Draw a diagonal of any quadrilateral and it splits into two triangles. Each triangle has 180 degrees, and the six angles together make up the four corners of the quadrilateral. So the angles of any quadrilateral add to 360 degrees. Three right angles use 270, leaving exactly 90 for the fourth.

Scenario:

A school is laying a new classroom floor and wants the base marked out as an exact rectangle. The mason has two long ropes of equal length and some pegs, but no large set square. He pegs the two ropes so that each crosses the other at its own midpoint, and then pegs the four rope ends and runs a string round them. A teacher watching asks three things. Will the figure be a rectangle whatever angle the ropes make? What would have to change to mark out a square instead? And would measuring the finished corners prove the method works?

Analysis:

The method rests on two facts about diagonals, and the third question is about the difference between checking a case and proving a rule.

Title:

Fold the three definitions

Safety:

Take care with the scissors, and cut away from your fingers.

What You Notice:

The four-right-angle shape already has equal opposite sides and equal bisecting diagonals, which is why the short definition is enough. The square adds a right angle at the crossing.

Carpenters in Europe square a frame by checking that its two diagonals are equal, and farmers in Mozambique use the same trick to set out the rectangular base of a house.

Measuring a thousand rectangles would still not prove that diagonals bisect each other. Only the congruence argument settles the thousand-and-first.

Equal and crossing at the middle — Two strips of equal length joined at their midpoints give a rectangle whatever angle they make. Set that angle to a right angle and you get a square.

Keeping one property and dropping another

A rectangle has parallel opposite sides. Are there quadrilaterals with parallel opposite sides that are not rectangles? Draw two pairs of parallel lines set at some angle other than a right angle. The shape in the middle has parallel opposite sides and no right angle anywhere.

Parallelogram

Such a shape is called a parallelogram. A parallelogram is a quadrilateral whose opposite sides are parallel. A rectangle has that property, so every rectangle is a parallelogram. It is a parallelogram with the extra feature of four right angles.

Finding the angles

Draw a parallelogram with sides of 4 and 5 centimetres and 30 degrees between them. One side acts as a transversal across the other pair. Co-interior angles across a transversal add to 180, so the angle next to the 30 is 150.

Going round the shape

The same argument works at every corner. Each pair of adjacent angles adds to 180. So the four angles here are 30, 150, 30 and 150. The two angles of 30 are opposite each other, and so are the two of 150.

Does that always happen?

Call one angle x. Its neighbour is 180 less x, since the two add to 180. The next neighbour is 180 less that, which comes back to x. So opposite angles of a parallelogram always match, whatever x is.

What about the sides?

The opposite sides look equal. Draw a diagonal and compare the two triangles. The opposite angles of the parallelogram give one equal pair. The diagonal is a transversal across a pair of parallel sides, so alternate angles give a second equal pair.

So they are equal

The diagonal itself is common to both triangles. By the angle-angle-side condition the triangles are congruent, so the opposite sides must be equal. That is a property now, not a guess from looking.

Are the diagonals equal?

In a rectangle they were. Measure them in the parallelogram you drew and they will not match. The equal diagonals belonged to the right angles, and we have given those up. This is worth holding on to. Each shape in this chapter keeps some of the rectangle’s properties and loses others, and knowing which is which is most of the work.

Do the diagonals still bisect?

That part survives. Take the two triangles on either side of the crossing point. One pair of opposite sides is equal, and two pairs of alternate angles are equal because the sides are parallel. The angle-side-angle condition gives congruence.

Four properties of a parallelogram

Opposite sides are parallel. Opposite sides are equal. Adjacent angles add to 180 and opposite angles are equal. The diagonals bisect each other, though they need not be equal.

The other thing to drop

A square has right angles and equal sides. We dropped the right angles and kept parallel sides. What if we keep the equal sides instead? Draw two equal sides meeting at 50 degrees, then find the fourth corner at that same distance from both free ends.

Rhombus

A quadrilateral with all four sides the same length is called a rhombus. The one just drawn has an angle of 50 degrees, and any angle under 180 would have done. So a rhombus is far from being only a tilted square.

The four angles at a diagonal

In a rhombus, draw one diagonal. Two triangles appear, each with two equal sides, so each has equal base angles. The two triangles are also congruent to each other. Putting those together, all four angles the diagonal makes with the sides are equal.

Which makes it a parallelogram

Those equal angles are alternate angles across the diagonal. Equal alternate angles force the two lines to be parallel. The same holds for the other pair, so the opposite sides are parallel and every rhombus is a parallelogram.

So it inherits everything

Being a parallelogram, a rhombus has adjacent angles adding to 180, equal opposite angles, and diagonals that bisect each other. With one angle of 50 degrees, its angles are 50, 130, 50 and 130.

Two extras of its own

A diagonal splits each corner into two equal angles, so the diagonals of a rhombus cut its corners in half. And congruence at the crossing gives two equal angles on a straight line, so the diagonals cross at right angles. Note what is still missing: nothing here makes the two diagonals equal. Measure them in the rhombus you drew and they will differ widely.

Putting the four shapes together

Parallelograms form the widest set. Rectangles sit inside them, and so do rhombuses. A shape that is in both of those has right angles and equal sides, which makes it a square. So the squares are exactly the overlap.

Scenario:

A folding clothes airer is built from four metal rods of the same length. They are hinged into a ring, so the shape can be opened out or squeezed flat. When it is squeezed, the angle at each hinge changes. The rods, of course, stay the same length. The owner measures the two distances across the airer, corner to opposite corner. Both change as she folds it. They are never equal to each other, except at one setting. She wonders three things. What shape is the frame always? What is special about that one setting? And do the two cross distances always meet at right angles?

Analysis:

The rods never change length, so one property is fixed throughout, while the angles and the two cross distances are free to change together.

Title:

Sort the quadrilaterals

Safety:

Take care with the scissors, and cut away from your fingers.

What You Notice:

The square is the only one that goes in the overlap, because it alone has both right angles and equal sides. Nothing lies outside the large oval, since all four have parallel opposite sides.

A rhombus is a parallelogram without being told to be one. Equal sides make the alternate angles equal, and equal alternate angles are exactly what forces two lines to be parallel.

A shape can be a rectangle and a rhombus at the same time. When it is both, it is a square, which is why the square sits where the two sets overlap.

Drop the angles, keep the sides — A parallelogram is a rectangle that has given up its right angles. It keeps equal opposite sides and bisecting diagonals, and loses only the equal diagonals.

Predicting a shape from its diagonals

Take a geoboard, or a sheet of dotted paper, and stretch two rubber bands as diagonals. Make them equal in length and set them at right angles, each crossing the other at its midpoint. Join the four ends. What shape have you made?

The first answer

It is a square. Equal diagonals that bisect each other give four right angles, and the right angle at the crossing then makes the four sides equal. Both facts came from the first topic, so the shape can be named before it is drawn.

Change one thing

Now stretch one of the bands so that it sticks out 2 centimetres further at each end. The diagonals still cross at right angles at their midpoints, but they are no longer equal. The four sides stay equal, so the shape is a rhombus, and no longer a square.

Building from a triangle instead

Here is another way to make quadrilaterals. Cut out two identical triangles and join them along a side of equal length. There are two genuinely different ways to do it, and they give different shapes.

The flip

Lay the second triangle as a mirror image of the first across the shared side. The two free sides of one copy now sit next to the matching sides of the other. So the quadrilateral has two pairs of EQUAL ADJACENT sides.

The turn

Instead, rotate the second triangle half a turn about the midpoint of the shared side. Now the sides pair up across the shape rather than beside each other, so the opposite sides are equal. That makes a parallelogram.

Two equilateral triangles

Take two equilateral triangles of side 8 centimetres and join them on a side. Every side of the result is 8 centimetres, so it is a rhombus. Its angles are 60 and 120 degrees, which is the diamond you see on tiled floors.

Two isosceles triangles

Now take two copies of an isosceles triangle with sides 8, 8 and 6 centimetres. Flip across the short side and the four sides are all 8, giving a rhombus again. Flip across a long side instead and the sides run 8, 6, 6, 8, which is a new shape.

Kite

That shape is called a kite. A kite is a quadrilateral whose corners can be labelled so that two pairs of adjacent sides are equal: one pair meeting at one corner and the other pair meeting at the opposite corner.

A scalene triangle gives six

A scalene triangle has three different sides, so it can be joined along any one of them, by a flip or by a turn. That is six quadrilaterals from a single triangle: three kites and three parallelograms.

What a kite’s diagonals do

Label a kite so that two sides meeting at one corner are equal, and the other two meet at the opposite corner. Draw the diagonal joining those two corners. It splits the kite into two triangles that are congruent by side-side-side, because both pairs of sides are equal and the diagonal is shared.

Reading off three properties

From that congruence, the diagonal bisects the angles at both of those corners. It also cuts the other diagonal exactly in half, and it meets it at right angles. Note what is NOT true: the diagonal being cut in half is not itself cut in half by the other.

Relaxing the parallel condition

A parallelogram needs both pairs of opposite sides parallel. What if we ask for only one pair? That gives a wider family, and its members are called trapeziums. A trapezium is a quadrilateral in which at least one pair of facing sides runs parallel.

So a parallelogram is one too

Read the definition carefully. Having two pairs of parallel sides certainly counts as having at least one pair. So every parallelogram is a trapezium, and every rectangle, rhombus and square is as well.

Finding the angles of a trapezium

The parallel pair does the work. Each of the other two sides is a transversal across them, so co-interior angles add to 180 degrees. The two angles at one end of a non-parallel side are supplementary, and so are the two at the other end.

Isosceles trapezium

When the two non-parallel sides of a trapezium happen to be equal in length, it is called an isosceles trapezium. Such a shape looks symmetric, and that appearance can be turned into a proof.

Proving the base angles equal

Drop a perpendicular from each end of the shorter parallel side down to the longer one. The piece between them is a rectangle, since it has parallel sides and right angles. So the two perpendiculars are equal in length.

Finishing the argument

That leaves two right-angled triangles, one at each end. Each has the same perpendicular height, and the slanting sides are equal because the trapezium is isosceles. So the triangles are congruent, which makes the two base angles equal.

Scenario:

A school is cutting shapes for a stage backdrop from two identical triangular offcuts of plywood. Each offcut has sides of 50, 50 and 30 centimetres. The art teacher wants two different panels from them. The first panel must have all four edges the same length. The second must have its opposite edges parallel, so that it can sit flush against a straight batten at the top and bottom. A pupil says the two panels will need four offcuts in total, because one pair of triangles can only make one shape.

Analysis:

Both panels come from the same pair of triangles. What changes is which edge they are joined along, and whether one is flipped or turned.

Title:

Prove the base angles yourself

Safety:

Take care with the scissors and the corner of the set square.

What You Notice:

The two end triangles sit exactly on top of each other, and the base angles measure the same. The middle piece is a rectangle, which is what makes the two heights equal.

A scalene triangle has three different sides, so it can be joined to its copy along any of them, by a flip or a turn. That gives six different quadrilaterals from a single triangle.

Every parallelogram is a trapezium under the usual definition, because having two pairs of parallel sides certainly counts as having at least one.

Flip for a kite, turn for a parallelogram — One triangle and its copy make two different quadrilaterals. Reflect across the shared side and adjacent sides pair up, giving a kite. Rotate about its midpoint and opposite sides pair up instead.