More Geometry

Patternity_ShapeSections.jpegThis 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 More_Geometry_01.gif
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°
More_Geometry_03.gif

4
Square
360⁄4 = 90°

180 − 90 = 90°
More_Geometry_05.gif

6
Hexagon
360⁄6 = 60°

180 − 60 = 120°
More_Geometry_07.gif

8
Octagon
360⁄8 = 45°

180 − 45 = 135°

 More_Geometry_09.gif
 

 

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.

More_Geometry_Parallel_01.gif More_Geometry_Parallel_02.gif More_Geometry_Parallel_03.gif More_Geometry_Paralle_04.gif
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.

More_Geometry_Parallel_05.gif
More_Geometry_Parallel_06.gif
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.

More_Geometry_Parallel_07.gif
More_Geometry_Parallel_08.gif
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
More_Geometry_Parallel_09.gif
More_Geometry_Parallel_10.gif
More_Geometry_Parallel_11.gif

x = 105o

Reason: Corresponding angles on parallel lines are equal.

y = 65o

Reason: Alternate angles on parallel lines are equal.

z + y = 180
z + 65 = 180
z = 115o

Reason: Angles on a staraight line add up to 180 degrees.

w + 75 = 180
w = 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
More_Geometry_Bearings_01.gif

 

Comparisons

 
More_Geometry_Bearings_02.gif

For an exciting and fun practice with bearings click here.
Compass Bearing
True Bearing
N
000o
NE
045o
E
090o
SE
135o
S
180o
SW
225o
W
270o
NW
315o