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.