UNIT 8 SEC 2.1 Triangles

UNIT 8 SEC 2.1 Triangles

29 April 2016

20:00

The interior angles of a triangle are the angles formed inside the triangle

by its sides. For example, the interior angles of the triangle in Figure 12

are marked as α, β and γ. You may be familiar with the fact that the

interior angles of every triangle add up to 180◦. In the next activity you

are asked to check this result for some triangles, and then prove it by using

results that you found earlier.

The angles in a triangle add up to 180◦.

First, a key step was experimenting with the triangle, spotting a pattern

and deciding what result seems to be true in general. Then, using results

that were already proven, and working step by step with a clear diagram,

it was possible to prove the result.

An addition to a geometric diagram, in order to help prove a fact about the

original shape, is known as a construction. A construction that is a line

is known as a construction line.

Once you have come up with the main ideas of a proof, the next stage is to

write the proof out clearly, using words and mathematical notation, so

that a reader can understand it. You must give an argument that refers to

results known to be true, and work logically step by step from what you

know to what you want to prove. The process of setting out a geometric

argument is summarised overleaf, and then Example 3 shows how it is

applied to prove that the angles in a triangle add up to 180◦.

Setting out a geometric argument

1. State the general fact that is to be proved, including any given

information.

2. Draw a diagram that contains the information that is given,

labelling the important features, such as points and angles.

3. Add any useful constructions.

4. Proceed step by step from what is given to what is to be proved,

explaining your reasoning clearly.

Example 3 Proving the result about the angles in a triangle

Prove that the angles of any triangle add up to 180◦.

Solution

State the general fact to be proved.

Here we prove that the angles in a triangle add up to 180◦.

Draw a suitably-labelled diagram containing the information that you

know.

Consider -ABC.

Add any necessary constructions.

Draw a line through B, parallel to the side AC.

Proceed step by step from what you know to the fact that is to be

proved. Explain each step clearly.

∠DBA and ∠BAC are alternate angles. Therefore ∠DBA = θ.

∠EBC and ∠BCA are alternate angles. Therefore ∠EBC = φ.

∠DBA, ∠ABC and ∠EBC are on the straight line passing through B and

so add up to 180◦.

So θ + ψ + φ = 180◦.

But this is the sum of the angles in -ABC.

Thus the angles in a triangle add up to 180.

You will be looking at some more proofs later in the unit, but in the rest of

this subsection you will see how the result about the sum of the angles in a

triangle can be used in different situations. First we look at two special

types of triangle: equilateral and isosceles triangles. ‘Isosceles’ is pronounced

‘eye-sos-eh-lees’.

If a triangle has all its sides the same length, then all its angles are equal

and the triangle is known as an equilateral triangle. On a geometric

diagram, you can show that two or more line segments have the same

length by putting a stroke, or the same number of strokes, on each of the

line segments, as shown in Figure 13.

Each angle of an equilateral triangle is 60◦.

If a triangle has just two sides that are the same length, then it also has

two equal angles, known as the base angles, and the triangle is called an

isosceles triangle. As shown in Figure 14, the third angle is known as

the apex angle.

A triangle in which one angle is equal to 90◦ is called a right-angled

triangle. Example 4 considers the angles in a right-angled triangle that is

also isosceles.

Example 4 Finding angles in an isosceles triangle

Calculate the base angles in an isosceles right-angled triangle, such as the

one shown below.

Solution

Start from what you know.

Let each base angle of the triangle be θ (the same letter can be used for

each angle since the angles are equal). Then the angles of the triangle are

90◦, θ and θ.

Using the fact that the angle sum of a triangle is 180◦ gives the equation

90

◦

+ 2θ = 180

Solve the equation.

Rearranging the equation gives

2θ = 180

◦− 90

2θ = 90

θ = 45

So each base angle of the triangle is 45◦.

If you can show that two angles in a triangle are equal to each other, then

you can deduce that the triangle is isosceles (or possibly equilateral) and

that the sides opposite these two angles have the same length.

A triangle that is neither equilateral nor isosceles is known as a scalene

triangle. All its sides are of different lengths.

The next example involves putting together several results that you have

met so far. You will see that the steps of drawing a diagram, adding

construction lines and working logically apply here too. The example

involves finding some unknown angles: each new angle is found using

information that was either known at the start or worked out earlier in the solution.

Example 5 Finding angles related to a garden shed

The diagram below shows the front of a garden shed, including a wooden

strip that needs to be replaced. A close-up of the replacement strip is

shown in the inset. The strip makes an angle of 65◦ with the vertical. The

lines AB and DC are parallel, and the lines AE and BC (being vertical)

are also parallel. Calculate the angles α, β, γ and δ.

20

Solution

Use the angle properties of straight lines, parallel lines and angles.

Since the angles on a straight line add up to 180◦,

γ = 180

◦− 65

= 115

Since the lines DC and AB are parallel, the angles marked β and 65◦ are

corresponding angles. So

β = 65

Draw construction lines to help you to find the unknown angles.

Since the lines AE and BC are parallel, ∠FAE and ∠ABC are

corresponding angles. So ∠FAE = β = 65◦.

Since the angles on a straight line add up to 180◦,

α = 180

◦− 65

= 115

Since the angles in -CGD add up to 180◦,

∠CDG = 180

◦− 90

◦− 65

= 25

Hence, since the angles on a straight line add up to 180◦,

δ = 180

◦− 25

= 155

Example 6 Finding an expression for an angle

Find an expression for ∠ABC in the triangle below in terms of α.

Solution

Since the sum of the angles in a triangle is 180◦,

∠ABC + 50

+ α = 180

Hence

∠ABC = 180

◦− 50

◦− α

so

∠ABC = 130

◦− α.

∠ABC can now be labelled as 130◦− α on the diagram.

Next, you are asked to use the dynamic geometry resource to discover

something about the sum of the exterior angles of a triangle. An exterior

angle of a triangle is the angle formed outside the triangle by one side and

an extension of the adjacent side,

UNIT 8 SEC 1,2 Pairs Of An Equal Triangle

UNIT 8 SEC 1,2 Pairs Of An Equal Triangle

28 April 2016

13:56

In a geometric diagram, parallel lines are indicated by putting matching

arrowheads on the lines. When a diagram contains two pairs of parallel

lines, one pair is marked with a single arrowhead on each line and the

other pair is marked with a double arrowhead on each line.

When you are looking at a geometric diagram, you should take care not to

assume properties that are not marked.

Opposite angles

Opposite angles are equal.

This is a useful result, as it means that as soon as you spot a pair of angles

that are opposite to each other, you can deduce that they are equal.

Corresponding and alternate angles

The angles α and β in Figure 8 are called corresponding angles, because

they are in corresponding positions on the two parallel lines

   Now let’s look at the angles formed when a line crosses a pair of parallel

lines,

The angles α and β in Figure 8 are called corresponding angles, because

they are in corresponding positions on the two parallel lines.

   These angles are equal, because if you slide angle α up then it lies exactly

on top of angle β, as you saw in Activity 4(b). Any corresponding angles

can be seen to be equal in the same way, which gives the important result

stated below.

Corresponding angles are equal.

One pair of corresponding angles is marked with single arcs, and a second pair is marked with double arcs. This is a convention used frequently in geometric

diagrams when angles are not labelled with individual letters or with their

sizes: equal angles are indicated by marking them with the same number

of arcs.

The two angles marked with double arcs are part of a capital F, and so

corresponding angles are also known informally as F angles.

 The angles α and γ are known as alternate

angles, because they are on alternate sides of the line that crosses the pair

of parallel lines. They are also known informally as Z angles, because

there is a pair of such angles in a capital Z.

Also, the angles β and γ are equal since they are

opposite angles, and hence the alternate angles α and γ are equal – you

saw this argument in Activity 4. A similar argument applies to other pairs

of alternate angles, so we have the following result.

Alternate angles are equal.

Not all pairs of alternate angles look like angles in a letter Z! For example,

in Figure 10 the two angles marked θ and φ are obtuse angles, but they are

alternate angles nevertheless. The angles marked ψ and ω are also

alternate angles.

Example 2 Finding corresponding and alternate angles

Calculate the angles α and β in the diagram below.

Solution

Look for alternate, corresponding and opposite angles.

The line segments AC and DE are parallel, so ∠ABD and ∠BDE are

alternate angles.

So α = 70◦.

Add a line segment to the diagram to help you spot equal angles.

Extend CE to a point F, as shown in the margin. Then since AC and DE

are parallel, ∠DEF and ∠BCE are corresponding angles. Since

∠BCE = 100◦, it follows that ∠DEF = 100◦.

Since β and ∠DEF are angles on a straight

When you work on a problem like that in Example 2, you will probably

find it helpful to mark the sizes of the angles on the diagram as you find

them.

Suppose that two lines are crossed by a third line, as

shown in Figure 11. The result (in terms of the diagram in Figure 11) is:

If the first two lines are parallel, then the angles α and β are equal.

This result also works in reverse, in the sense that what is known as the

converse result is true:

In general, the converse of the result ‘If A is true, then B is true’ is the

result ‘If B is true, then A is true’.

There is an important point here. A large proportion of mathematical

results are of the form ‘If A is true, then B is true’ – and it is not always

the case that the converse of a result is also a mathematical result.