Class 7 Mathematics Chapter 9: Geometric Twins

Two shapes that are really the same

Cut two triangles from card using the same template. Put one on top of the other and it disappears underneath.

Shapes that fit over each other exactly like this are called congruent. They are geometric twins.

The test is covering, not looking

Looking at two shapes side by side can mislead. The test is whether one can be laid over the other with no part sticking out and no gap.

If it can, they are congruent. If it cannot, they are not.

Turning and flipping are allowed

The second shape may need turning before it fits, or even flipping over. That is still allowed.

Congruence is about size and shape, not about which way round a shape happens to be lying.

Same shape is not enough

Two triangles can have exactly the same shape and still not be congruent. A small one and a large one of the same shape will never cover each other.

The angles agree, but one is simply bigger.

Same size is not enough either

Two shapes can also cover the same amount of paper without being congruent. A long thin rectangle and a square can have equal areas.

Neither one would lie on top of the other at all.

Congruent means both at once

So congruence demands the same shape AND the same size. Nothing less will let one shape cover the other.

That is the whole of it, and it is why the covering test settles the question.

Which corner matches which?

Once two triangles are congruent, there is more to say. Each corner of one sits over a particular corner of the other.

Those are called corresponding vertices, and the sides and angles at them correspond too.

The correspondence must be stated

Saying that two triangles are congruent is not the whole message. Which corner goes to which corner matters just as much.

Without it, nobody knows which side of one equals which side of the other.

How it is written

We write that triangle ABC is congruent to triangle PQR. The ORDER of the letters carries the correspondence.

A goes to P, B goes to Q, and C goes to R. That is the agreement the notation makes.

Reading the notation

So the statement tells you where every part goes. Side AB corresponds to side PQ, because A goes to P and B goes to Q.

The angle at A corresponds to the angle at P. Every part can be read off in this way.

Six facts from one statement

A congruence statement between two triangles gives six equalities at once. Three sides are equal and three angles are equal.

That is why the statement is worth writing carefully. One line of notation replaces six separate measurements.

It is also why the notation is worth trusting. Once the correspondence is known to be right, no part of either triangle needs measuring again.

A worked pair

Suppose triangle ABC has sides AB of 4 cm, BC of 6 cm and CA of 7 cm, and that triangle ABC is congruent to triangle PQR.

Then PQ is 4 cm, QR is 6 cm and RP is 7 cm. No measuring was needed, because the notation said which side was which.

The order is not free

Now suppose somebody wrote instead that triangle ABC is congruent to triangle QRP. That is a different claim.

It would make PQ equal to CA, which is 7 cm rather than 4 cm. One of the two statements must be wrong.

Checking a stated correspondence

So a correspondence can be checked. Read off what it claims and compare that with the measurements.

If it says a 4 cm side matches a 7 cm side, the correspondence is wrong even if the two triangles really are congruent.

Angles work the same way

Take a second pair, with angles of 40, 60 and 80 degrees. If the correspondence is stated, each angle of one is matched to a definite angle of the other.

A correspondence that matched the 40 degree angle to the 80 degree one would be wrong, however congruent the triangles were.

What carries forward

Congruent shapes fit over each other exactly, after any turning or flipping needed. Same shape alone is not enough and same size alone is not enough: it takes both.

A congruence statement carries a correspondence in the order of its letters, and that order gives six equalities, three of sides and three of angles. The order is not free, and a wrong order makes the statement false even for genuinely congruent triangles.

Scenario:

A workshop cuts triangular brackets from one template, so every bracket should be congruent to every other. An apprentice measures one bracket and finds its sides to be 4 cm, 6 cm and 7 cm. He labels it ABC with AB of 4 cm, BC of 6 cm and CA of 7 cm. He labels a second bracket PQR and writes that ABC is congruent to QRP. A supervisor says the brackets are certainly congruent but the statement is wrong. The apprentice replies that the order of letters cannot matter, since both brackets came off the same template.

Analysis:

Two separate questions are being run together: whether the brackets are twins, and which corner of one sits over which corner of the other.

Activity:

Cut six congruent card triangles, label each one differently, and write a correct congruence statement for every pair.

Title:

Cut, cover and label

Safety:

Take the usual care with the scissors. No other hazards.

What You Notice:

The template pair fits exactly however it is turned or flipped. The square and the rectangle never fit, although their areas match. The reordered statement claims at least one pair of unequal sides to be equal, which is what makes it false.

One congruence statement replaces six separate measurements, three of sides and three of angles.

A correspondence can be false even when the two triangles really are congruent, because the letters carry a claim of their own.

Flipping a shape over is allowed when testing congruence, since congruence is about size and shape rather than position.

Congruent means same shape and same size.

The letter order is the correspondence.

Must you measure all six?

Two triangles have six measurements each: three sides and three angles. To prove them congruent, must all six pairs be checked?

They need not. Three well-chosen measurements are usually enough, and that is what saves the work.

Three sides are enough

Give the three sides as 5 cm, 6 cm and 9 cm. Any triangle with those three sides is congruent to any other.

This is the side-side-side criterion, written SSS for short.

Why three sides settle it

We have met the reason already. Draw the longest side, then sweep an arc from each end with the other two lengths.

The arcs cross in one place, so the third corner has nowhere else to go. The triangle was never free to differ.

Two sides and the angle between

Now give two sides of 8 cm and 11 cm with an angle of 35 degrees BETWEEN them. Again only one triangle fits.

This is side-angle-side, written SAS.

The angle has to be the included one

The word between is doing real work there. Draw the 35 degree angle, mark 8 cm along one arm and 11 cm along the other, and join the marks.

Every step was forced. The included angle holds the two sides apart and so fixes the third side.

Two sides and some other angle may fail

Now move the angle. Suppose we are given sides of 5 cm and 8 cm, and an angle of 30 degrees that is NOT between them.

Two different triangles can be built from that. The second side can swing to meet the base in two separate places, giving two shapes that are not congruent.

So SSA is not a criterion

Because two different triangles fit, the data has not settled anything. Side side angle is not a congruence criterion at all.

The lesson is that SAS needs the angle in the right place, not just somewhere.

Two angles and the side between

Give two angles of 25 and 95 degrees with a side of 10 cm BETWEEN them. Only one triangle fits.

This is angle-side-angle, written ASA.

Why ASA works

Draw the 10 cm side, then set 25 degrees at one end and 95 degrees at the other. The two angles use 120 degrees, so the third is 60.

Since 120 is under 180, the arms lean towards each other and must cross. They cross in one place only.

Right triangles get their own criterion

A right-angled triangle has a longest side opposite the right angle, called the hypotenuse.

Give the right angle, a hypotenuse of 13 cm and one other side of 5 cm. That fixes the triangle completely.

What RHS says

This criterion is right angle, hypotenuse and side, written RHS. The angle named is always the right angle.

It is worth having separately, because the right angle is not between the two named sides, so SAS does not cover it.

Three angles are never enough

Give three angles of 30, 70 and 80 degrees and nothing else. Many triangles fit.

Angle-angle-angle is not a criterion. It never was, and it never can be.

Why AAA fails

Angles fix the shape of a triangle and say nothing about its size. A small triangle and a large one can carry exactly those three angles.

So at least one length is always needed, and that is why every criterion includes a side.

Reading a question for its criterion

So the useful habit is to ask which measurements you have been given, and WHERE the angle sits.

Three sides is SSS. Two sides with the angle between is SAS. Two angles with the side between is ASA. A right angle with the hypotenuse and a side is RHS.

Naming it is part of the answer

In a proof, the criterion should be named. Saying that two triangles are congruent by SAS tells a reader which three facts were used and why they are enough.

Without the name, a correct answer looks like a guess.

It also guards against the two failures above. Writing down which criterion you are using forces you to notice when the data does not match any of them.

What carries forward

Three measurements are usually enough, and four criteria do the work: SSS, SAS, ASA and RHS. Each of them includes at least one side, because angles alone fix only the shape.

Two combinations are not criteria. Three angles leave the size open, and two sides with an angle outside them can give two different triangles. In a proof, name the criterion you are using.

Scenario:

A surveyor must show four pairs of triangular plots congruent. For the first pair she has measured all three sides of each, getting 5 m, 6 m and 9 m both times. For the second she has two sides of 8 m and 11 m with the 35 degree angle between them. For the third she has angles of 25 and 95 degrees with the 10 m side between them. For the fourth she has only three angles, 30, 70 and 80 degrees, and says that three matching measurements must be enough. A colleague adds that a fifth pair, with sides 5 m and 8 m and a 30 degree angle outside them, is also settled.

Analysis:

Three of the claims hold under a named criterion, and two of them rest on combinations that are not criteria at all.

Activity:

Collect five pairs of triangle measurements, name the criterion for each that is settled, and say what is missing from the rest.

Title:

Build them and see

Safety:

Take care with the compass point and the protractor. No other hazards.

What You Notice:

The first three sets each give one triangle however often you repeat them, which is SSS, SAS and ASA. The fourth gives two different triangles, and the last gives as many as you care to draw, which is why neither is a criterion.

Three measurements prove congruence, but not any three: three angles never will, however carefully they are measured.

Two sides and an angle outside them can be built into two different triangles, which is why SAS insists on the included angle.

RHS earns its own place because the right angle does not sit between the hypotenuse and the other named side.

SSS, SAS, ASA, RHS: every one names a side.

In SAS the angle sits between.

A triangle with two equal sides

Cut a triangle from paper with two of its sides the same length. This is an isosceles triangle.

The corner between the two equal sides is called the apex. The third side is the base, and the two corners at its ends are the base angles.

Fold it and see

Now fold the triangle so that one equal side lands exactly on the other. The crease runs from the apex down to the base.

The two halves land on top of each other. That tells us something, and it is worth saying exactly what.

The two halves are congruent

The crease meets the base at its midpoint, so each half has half the base. Each half also has one of the equal sides, and they share the crease itself.

So the two halves have three pairs of equal sides. By SSS they are congruent.

Which gives the base angles

Congruent triangles have equal angles in corresponding places. One base angle sits in each half, in the same position.

So the two base angles of an isosceles triangle are equal. The fold has proved it, not merely suggested it.

Finding the base angles from the apex

Now we can calculate. Suppose the apex angle is 72 degrees. The three angles must total 180, so the two base angles share 108 between them.

They are equal, so each one is 54 degrees.

A second case

Suppose instead the apex angle is 100 degrees. Then the base angles share 80 degrees between them.

Each one is 40 degrees. Notice this triangle is obtuse at the apex and still perfectly isosceles.

Going the other way

The calculation runs backwards too. Suppose you are told the base angles are 54 degrees each.

Together they use 108, so the apex angle is 72 degrees. One base angle is enough to find everything.

An equilateral triangle

Now take a triangle with all three sides equal. This is an equilateral triangle.

Any one of its three corners can be treated as the apex, because every pair of sides is a pair of equals.

All three angles are 60

Choose one corner as the apex. The other two angles are base angles, so they are equal.

Choose a different corner and the same argument makes another pair equal. So all three angles are equal, and three equal angles totalling 180 are 60 each.

No measuring was needed

Notice that nothing in that argument mentioned a length. It works for a triangle of side 2 cm and for one of side 2 metres.

That is why every equilateral triangle has 60 degree angles, whatever its size.

Equilateral is a special isosceles

An isosceles triangle needs at least two equal sides, and an equilateral triangle has three. So every equilateral triangle is isosceles as well.

The reverse does not hold. A triangle with apex 100 and base angles 40 is isosceles and certainly not equilateral.

The converse is true as well

So equal sides give equal angles. What about the other direction?

If two angles of a triangle are equal, the sides facing them are equal too. So two equal angles are enough to prove a triangle isosceles.

Using the converse

That is useful in practice. Suppose a triangle has angles of 40, 40 and 100 degrees, and no side has been measured at all.

The two 40 degree angles tell you that the sides facing them are equal, so the triangle is isosceles.

Which angle faces which side

Take care over the facing. In a triangle the side facing an angle is the one opposite it, not either of the two touching it.

So the equal sides in that example are the two that face the 40 degree angles, and the base is the side facing the 100 degree angle.

A check worth making

Whichever direction you work in, add the three angles at the end. They must come to 180.

For the first case, 72 and 54 and 54 add to 180. For the second, 100 and 40 and 40 add to 180 as well.

The check is worth the few seconds. It catches the commonest slip here, which is halving the apex angle instead of halving what is left over.

What carries forward

Folding an isosceles triangle makes two halves that are congruent by SSS, which proves the base angles equal. From the apex angle the base angles follow, and from a base angle the apex follows.

Every equilateral triangle has three 60 degree angles, at any size, and is isosceles too. The converse also holds: two equal angles mean the sides facing them are equal.

Scenario:

A builder makes a roof gable as an isosceles triangle with an apex of 72 degrees. He needs both base angles. A second gable has an apex of 100 degrees. An apprentice says that one cannot be isosceles, because the angle is obtuse. A third gable is specified only by its base angles, 54 degrees each, and the builder needs the apex. The apprentice then measures a decorative tile at 40, 40 and 100 degrees. He says nothing can be said about its sides until they are measured.

Analysis:

Two calculations run in opposite directions, one objection rests on a familiar picture, and one claim ignores the converse.

Activity:

Measure the angles of five triangular objects, and for each one decide from the angles alone whether its sides include an equal pair.

Title:

Fold, measure and reason

Safety:

Take the usual care with scissors and the protractor point. No other hazards.

What You Notice:

The crease lands at the midpoint and the base angles come out equal, 54 degrees in the first case and 40 in the second. The equilateral triangle gives 60 degrees from every fold, and in the last triangle the two sides facing the 40 degree angles are the matching pair.

An isosceles triangle can be obtuse, as long as the obtuse angle sits at the apex rather than at the base.

Any one angle of an isosceles triangle is enough to find all three, whichever angle you are given.

Two equal angles prove a triangle isosceles without a single side being measured.

Subtract from 180 first, then halve.

A side faces the angle opposite it.