Angles, Parallel Lines and Bearings

 

Angles In Par­al­lel Lines

We use arrow­heads to indi­cate that 2 lines are parallel:-

Arrowheads to indicate parallel lines

 

 

 

 

 

When a line crosses 2 par­al­lel lines it cre­ates pairs of equal angles

Inte­rior Angles

In the dia­gram below x and y are inte­rior angles.

Inte­rior angles add up to 180°

so x + y = 180°

Interior angles add up to 180 degrees

 

 

 

 

 

Here’s a video that explains inte­rior (or co-interior angles) very clearly:-

Cor­re­spond­ing Angles

What are cor­re­spond­ing angles? That’s best explained by a cou­ple of exam­ples. In the fol­low­ing dia­grams; a and b and c and d are cor­re­spond­ing angles. Note that the lines that join up cor­re­spond­ing angles make an ‘F’ shape:-

Corresponding angles are equal

 

 

 

 

 

Here’s a video that explains cor­re­spond­ing angles clearly and in detail:-

Alter­nate Angles

This dia­grams shows 2 pairs of alter­nate angles.

Alter­nate angles are equal. Notice how the lines make a ‘Z’ shape.

Alternate angles on parallel lines are equal

Here’s a video to explain alter­nate angles in par­al­lel lines:-

Bear­ings

Bear­ings are mea­sured clock­wise from the North:-

Bearings- example of 60 degrees

 

 

 

 

 

 

To get a fuller under­stand­ing of all bear­ings, con­sider the com­pass. Can you work out each direc­tion shown here as a bearing?

Compass bearings

 

 

 

 

 

 

 

 

NE = 045

E = 090

SE = 135

S = 180

SW = 225

W = 270

NW = 315

Here’s a video that explains bear­ings in some detail:-

Here’s another video that walks through some Bear­ings exam questions:-

Par­el­lel Lines and Bearings

You can use the fact that North lines are always par­al­lel and knowl­edge of angle facts to work out bearings.

This is best illus­trated with an exam­ple. In the fol­low­ing dia­gram, work out the bear­ing of B from A.

Bearings question using North , parallel lines and angle rules

 

 

 

 

 

 

 

 

There are 2 pos­si­ble approaches to answer­ing this ques­tion. Both start with draw­ing the North line from A so that we have 2 par­al­lel lines point­ing North.

Draw the parallel North line

 

 

 

 

 

 

 

 

The first method we can use is based on the fact that co-interior angles add up to 180°

There­fore we know the angle below marked in red must be 180° — 135° = 45°

Interior Angles add up to 180 degrees

 

 

 

 

 

 

 

 

But remem­ber bear­ings are mea­sured clock­wise from the North. So the bear­ing of B from A = 360 — 45 = 315 see the dia­gram below:-

Co-Interior Angle Solution to Bearing Question

 

 

 

 

 

 

 

 

 

Here’s an alter­nate solu­tion based on.….alternate angles. Again we start by draw­ing the North line from A:-

Draw the parallel North line

 

 

 

 

 

 

 

 

Then we find the alter­nate angles (it helps to extend the line below A and high­light the ‘Z’ shape we’re look­ing for):-

Deduce Bearings using alternate angles

 

 

 

 

 

 

 

 

 

 

 

Finally, remem­ber­ing that bear­ings are always mea­sured clock­wise from the North we just have to cal­cu­late the answer:-

Use alternate angles to deduce bearing- answer

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