## More Geometry

This topic contains material that is usually studied in later years, but continues on from the topics covered previously.

### Angles in Polygons

The exterior (outside) angles of any polygon always add up to 360°.

 a + b + c + d = 360o
The interior (inside) angles of a polygon have different sums depending on the number of sides in the polygon.
Regular Polygons
Regular polyons have equal sides and equal angles.
The table summarises the names and angles of some of the more common regular polygons.

 Number of sides Name Size of each exterior angle Size of each interior angle Diagram 3 Equilateral triangle 360⁄3 = 120° 180 − 120 = 60° 4 Square 360⁄4 = 90° 180 − 90 = 90° 6 Hexagon 360⁄6 = 60° 180 − 60 = 120° 8 Octagon 360⁄8 = 45° 180 − 45 = 135°

### Parallel Lines

When two lines are parallel the angles made by a third line crossing them have certain properties.

Corresponding angles

Corresponding angles on parallel lines are equal.
The following diagram shows the four possible pairs of corresponding angles.

 a = b c = d e = f g = h

Alternate angles

Alternate angles on parallel lines are equal.
The following diagram shows the two possible pairs of alternate angles.

 k = m n = p

Co-interior angles

Co-interior angles on parallel lines are add up to 180o (supplementary angles).
The following diagram shows the two possible pairs of co-interior angles.

 r = s v = w

Using these angle properties, the size of angles on parallel lines can be found.

Examples

 Example One Example Two Example Three x = 105o Reason: Corresponding angles on parallel lines are equal. y = 65o Reason: Alternate angles on parallel lines are equal. z + y = 180z + 65 = 180z = 115o Reason: Angles on a staraight line add up to 180 degrees. w + 75 = 180w = 105o Reason: Co-interior angles on parallel lines add up to 180o t = 105o Reason: Vertically opposite angles are equal.

### Bearings

Compass bearings
Compass bearings have a maximum of 90o and each one begins with either N or S, followed by the number of degrees east or west of that direction.

e.g. N40oW is shown in the diagram.

True or Whole Circle bearings
These bearings use North as 0o and the angle is measured in a clockwise direction.

e.g. The true bearing shown in the diagram is written as 320o.

True bearings are usually written with THREE digits. e.g. 085o

Comparisons