Class 7 Mathematics Chapter 7: A Tale of Three Intersecting Lines

Three lines, and what they enclose

Take three straight lines and let each pair cross. If the three crossings are three different points, the lines enclose a triangle.

So a triangle is what three intersecting lines leave behind. Three sides, three corners, three angles.

The most even triangle of all

A triangle with all three sides the same length is called equilateral. Equal sides, equal lengths all round.

Draw one with every side 5 centimetres and look at its corners. All three angles are the same as well.

Why its angles must all be equal

Nothing distinguishes one corner of an equilateral triangle from another. Turn it, flip it, and it looks the same.

If one angle were larger than another, turning the triangle would change the picture. It does not, so the angles are equal.

Working out what they are

The three angles of any triangle add to 180 degrees. In an equilateral triangle they are also equal.

Three equal angles adding to 180 are 60 degrees each. So every equilateral triangle has three 60 degree angles, whatever its size.

Size does not change the angles

A tiny equilateral triangle and a huge one both have three 60 degree angles. The sides grow and the corners do not change at all.

That is worth noticing. Some things about a shape depend on its size and some do not.

Building one from its sides

To draw a triangle of sides 3, 4 and 5 centimetres, start by drawing the longest side.

Then open a compass to 3 centimetres and sweep an arc from one end. Open it to 4 and sweep from the other. Where the arcs cross is the third corner.

Why the arcs find the corner

Every point on the first arc is exactly 3 centimetres from one end. Every point on the second is exactly 4 from the other.

So a crossing point is both at once, which is precisely what the third corner has to be.

Three lengths that refuse

Now try sides 2, 3 and 6 centimetres. Draw the 6 centimetre side and sweep arcs of 2 and 3 from its ends.

The arcs never meet. The two short sides together reach only 5 centimetres, which cannot span a gap of 6.

The rule behind the refusal

Two sides of a triangle must together be longer than the third. If they are not, they cannot reach from one end of it to the other.

Check 2, 3 and 6: the 2 and the 3 make 5, and 5 is less than 6. So no triangle exists.

Checking all three pairs

To be safe you have to check every pair against the remaining side. For 3, 4 and 5 the pairs give 7, 8 and 9, each longer than the side left over.

In practice the only pair that can fail is the two shortest, so checking those first settles it fastest.

Another set that fails

Try 4, 4 and 9. The two fours make 8, which is less than 9.

So these lengths cannot make a triangle either, however carefully you draw. No amount of care can fix an arithmetic obstacle.

The exact borderline

Now try 2, 3 and 5. The two short sides make exactly 5, the same as the third side.

The arcs just touch, on the long side itself. The three points lie in a straight line, so there is no triangle, only a flat line. Equality fails too.

Why equality is a failure

A triangle needs its third corner off the line joining the other two. If the two short sides only just reach, the corner lands on that line.

So the rule needs the word longer, not longer or equal. The borderline belongs with the failures.

What the rule is really saying

A straight line is the shortest route between two points. Going via the third corner is a detour.

A detour cannot be shorter than the direct route, which is exactly the statement that two sides together exceed the third.

Deciding before drawing

So three lengths can be tested by arithmetic alone, before any compass comes out.

Add the two shortest. If the total beats the longest, a triangle exists. If it equals or falls short, none does.

What carries forward

A triangle is what three intersecting lines enclose. An equilateral triangle has three 60 degree angles, at every size.

Three lengths make a triangle exactly when the two shortest together exceed the longest. Equality is a failure, because the third corner would land on the line.

Scenario:

A workshop keeps steel rods cut to whole numbers of centimetres. An apprentice is asked to braze three rods into a rigid triangle for a bracket. He first picks rods of 3, 4 and 5 centimetres and succeeds easily. He then picks 4, 4 and 9 and spends twenty minutes failing, blaming his clamp and then his compass marks. He next picks 2, 3 and 5 and reports that the three rods lie flat against each other but will not stand up as a shape. Finally he asks whether a 5 centimetre equilateral bracket would have different corner angles from a 50 centimetre one.

Analysis:

Two of his sets fail for arithmetic reasons, one of them on the exact borderline, and his last question is about what size does and does not change.

Activity:

Cut six paper strips of different whole-centimetre lengths, pick sets of three, and predict by arithmetic which will make a triangle before trying.

Title:

Predict, then build

Safety:

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

What You Notice:

The predictions come out right every time, and the 2, 3, 5 set is the interesting one: the strips reach but lie in a row, so nothing is enclosed. Both equilateral triangles measure 60 degrees at every corner, however different their sides.

An equilateral triangle has three 60 degree angles whether its sides are 5 centimetres or 50, because the argument never mentions length.

Three lengths can be tested for a triangle by arithmetic alone, before a compass is opened at all.

Only the two shortest sides can ever fail the test, so one addition settles any set of three lengths.

Add the two shortest and beat the longest.

Reaching exactly is still a failure.

The three angles always make 180

Draw any triangle at all and measure its three angles. They add to 180 degrees.

Draw a different one, of any shape or size, and they still add to 180. This is the single most useful fact about triangles.

Seeing why, with scissors

Draw a triangle on paper and tear off its three corners. Now lay the three torn corners side by side, with their points touching.

The three of them fit along a straight line exactly. A straight line is 180 degrees, so the three angles make 180.

Seeing why, with a parallel line

Here is the same argument without tearing. Draw a line through one corner, parallel to the opposite side.

The two angles either side of that corner are alternate angles with two of the triangle's angles, so they are equal to them. Together with the third they lie along a straight line.

Two angles give the third

So a triangle's angles are not independent. Fix two of them and the third has no choice.

If two angles are 50 and 60 degrees, they use 110 of the 180. The third must be 70 degrees.

That is worth checking against

It also gives a check on any measurement. Measure all three and add them. If the total is not 180, at least one measurement is wrong.

The check costs one addition and catches a slip that a careful drawing never would.

What is enough to fix a triangle

Three side lengths fix a triangle completely, as the arcs showed. But sides are not the only data that works.

The question worth asking of any data is whether exactly one triangle fits it, many do, or none does.

Two sides and the angle between

Give two sides of 6 and 7 centimetres and the angle of 50 degrees BETWEEN them. Exactly one triangle fits.

Draw the 50 degree angle first, mark 6 along one arm and 7 along the other, then join the two marks. Nothing was left to choose.

Why the angle must be the included one

The included angle is the one that holds the two sides apart. Change it and the far ends move closer or further.

So the included angle fixes the third side, and with it the whole triangle.

Two angles and the side between

Now give two angles of 50 and 60 degrees and the side of 7 centimetres BETWEEN them. Again exactly one triangle fits.

Draw the 7 centimetre side, then the 50 degree angle at one end and the 60 degree at the other. The two arms must meet, and where they meet is the third corner.

Why the two arms must meet

The two angles use 110 degrees between them, leaving 70 for the third. Since 110 is less than 180, the arms lean towards each other.

Two lines leaning towards each other must cross. So the triangle closes, and it closes in only one place.

What does not fix a triangle

Give three angles of 50, 60 and 70 degrees and nothing else. Many triangles fit.

A small one and a large one both have those angles. Angles fix the shape and say nothing about the size.

Why that is worth knowing

It means angle data alone can never settle a length. A builder told only the angles of a roof truss cannot cut a single beam.

At least one length is always needed, and exactly one is usually enough.

The exterior angle

Extend one side of a triangle past a corner. The angle between that extension and the next side is called an exterior angle.

At the 70 degree corner, the exterior angle is 180 minus 70, which is 110 degrees.

A neater way to get it

Notice that 110 is also 50 plus 60, the two angles at the other corners.

That is no accident. An exterior angle equals the sum of the two interior angles opposite it, always.

Why that follows

The exterior angle and its own interior angle sit along a straight line, so together they make 180.

The three interior angles also make 180. Take the shared interior angle away from both and what is left must be equal.

Which is quicker in practice

To find an exterior angle you can subtract from 180 or add the two opposite angles. Both give the same number.

Adding the two opposite angles is usually quicker, because those are often the angles you were given.

What carries forward

The three angles of any triangle make 180 degrees, which gives both the third angle from two and a check on any measurement.

Two sides with the angle between them, or two angles with the side between them, fix exactly one triangle. Three angles alone fix only the shape. An exterior angle equals the two interior angles opposite it.

Scenario:

A carpenter receives a drawing for a roof truss. It is marked with three angles, 50 degrees, 60 degrees and 70 degrees, and no lengths at all. He telephones the architect to say he cannot cut a single beam from it. The architect replies that three angles describe the triangle completely. The carpenter then asks for one length and is given the beam between the fifty and sixty degree corners as 7 metres. Later he measures a finished truss and gets angles of 50, 60 and 75 degrees, and wants to know whether that is possible. He also needs the angle between the extended bottom beam and the sloping beam at the 70 degree corner.

Analysis:

The architect has confused fixing a shape with fixing a triangle, and the last two questions are the angle-sum check and the exterior angle.

Activity:

Draw three triangles with angles 50, 60 and 70 at three different sizes, then measure the exterior angle at each 70 degree corner.

Title:

Tear, build, and check

Safety:

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

What You Notice:

Every angle set totals 180, and the two constructions each give one triangle with no choices left. The exterior angle comes out the same whether measured, subtracted from 180, or found by adding the two opposite angles.

Adding three measured angles costs one addition and catches a slip that the neatest drawing would never reveal.

A carpenter handed a truss drawing marked only 50, 60 and 70 degrees cannot cut a single beam from it, because no length follows.

An exterior angle can be found by subtracting from 180 or by adding the two opposite angles, and adding is usually quicker because those are the angles you were given.

Two angles given, the third is decided.

An exterior angle is the far two added.

Two separate questions about a triangle

A triangle can be sorted by its sides or by its angles. These are two different questions, and a triangle gets one answer to each.

So every triangle carries two labels, and the pair of them says a good deal about its shape.

Sorting by sides

All three sides equal makes it equilateral. Exactly two equal makes it isosceles. All three different makes it scalene.

Those three cases cover every possibility, because three lengths either all agree, agree in one pair, or all differ.

Sorting by angles

All three angles less than 90 degrees makes it acute. One angle of exactly 90 makes it right-angled. One angle more than 90 makes it obtuse.

Notice that each of these says ONE angle, not two.

Why only one big angle is possible

Suppose a triangle had two angles of 90 degrees or more. Those two alone would already use 180 degrees or more.

That leaves nothing, or less than nothing, for the third angle. So no triangle has two right angles, and none has two obtuse angles.

Equal sides mean equal angles

In an isosceles triangle the two equal sides face two equal angles. The third angle can be anything the sum allows.

A triangle with angles 80, 50 and 50 is isosceles and acute. One with 90, 45 and 45 is isosceles and right-angled.

Checking those two sets

For the first, 50 and 50 and 80 add to 180, and all three are under 90, so acute is right.

For the second, 45 and 45 and 90 add to 180 as well, and exactly one angle is 90, so right-angled is right.

Which pairs of labels can meet

Most pairs of labels do occur. A scalene triangle can be acute, right-angled or obtuse: 90, 60 and 30 is right-angled and scalene, while 110, 40 and 30 is obtuse and scalene.

An isosceles triangle can be any of the three as well, since 110, 35 and 35 is obtuse and isosceles.

The one pair that cannot

An equilateral triangle has three equal angles adding to 180, so each is 60 degrees and nothing can change that.

Sixty is less than 90, so an equilateral triangle is always acute. There is no right-angled or obtuse equilateral triangle at all.

Two special lines inside a triangle

From any corner you can draw two quite different lines to the opposite side. Both are useful, and they are easy to confuse.

The difference is in what each one is asked to do when it arrives.

The median

A median goes from a corner to the MIDDLE of the opposite side. It does not care about angles at all.

Every triangle has three medians, one from each corner, and all three pass through a single point inside.

That meeting point is worth noticing. Three lines drawn for quite separate reasons agree on one place, in every triangle you try.

The altitude

An altitude goes from a corner and meets the opposite side at a RIGHT ANGLE. It does not care where it lands.

It is the height of the triangle when that side is taken as the base, which is what makes it useful for area.

When the two coincide

In most triangles the median and the altitude from a corner are different lines. In an isosceles triangle the two from the apex are the same line.

The apex is the corner between the two equal sides, and that line is a mirror line for the whole triangle.

An altitude can fall outside

In an obtuse triangle the altitudes from the two sharp corners land outside the triangle, on the opposite side extended.

With angles of 110, 40 and 30 the obtuse corner leans so far over that the foot of the altitude has nowhere inside to land.

Why the median never does

A median always ends at the midpoint of a real side, and a midpoint lies on the side itself.

So all three medians stay inside the triangle, in every triangle, whatever its shape.

What carries forward

A triangle is named twice over, once by its sides and once by its angles. Only one angle can be 90 degrees or more, so each angle label names one angle.

Every pair of labels occurs except the right-angled or obtuse equilateral, which is impossible because all three of its angles are 60. A median goes to the midpoint of a side and stays inside; an altitude arrives at a right angle and may land outside.

Scenario:

A carpentry workshop keeps its triangular offcuts in labelled trays. One tray is marked acute isosceles, one right-angled scalene, one obtuse isosceles, and one obtuse equilateral. An apprentice measures four pieces. The first has angles of 80, 50 and 50 degrees. The second has 90, 60 and 30 degrees. The third has 110, 35 and 35 degrees. The fourth has 110, 40 and 30 degrees. She also needs to mark the height on one piece for an area calculation. She is unsure which line to draw on the obtuse piece.

Analysis:

Three measured pieces match trays, one tray can never be filled, and one measured piece has no tray at all.

Activity:

Measure six triangular offcuts, give each its two labels, and keep a tally of which pairs of labels actually turn up.

Title:

Name them, then draw the inner lines

Safety:

Take care with the protractor point and the set square corner. No other hazards.

What You Notice:

Each triangle gets one side label and one angle label. The equilateral attempt fails, because fixing all three sides equal always gives 60 degree angles. The median and altitude from one corner are two different lines, and on the obtuse triangle both sharp-corner altitudes land outside.

An equilateral triangle can never be right-angled or obtuse, because its three equal angles are forced to be 60 degrees each.

In an obtuse triangle two of the three altitudes land outside the triangle altogether, on the opposite side extended.

From the apex of an isosceles triangle the median and the altitude are the same line, and that line is a mirror line for the whole shape.

A median aims for the middle, an altitude for a right angle.

Equilateral means sixty, and sixty is always acute.