UNIT 8 SEC 1.1 Angles and lines

1.1     Angles and lines 

Thithis subsection introduces some terminology and notation that are useful for explaining geometric ideas clearly and concisely.

Let’s start with some definitions

. In geometry, a point has a position but no size. For example, the place where two lines cross (or two line segments meet) is a point.

A line is a straight line, as normally understood, but one that extends In MU123 you may assume that any line

where a point along its length is identified. 

For example, in Figure 1, assume that ACE and BCD are both straight lines. infinitely far in both directions.

 A finite portion of a line, which is all that you can draw in practice, is called a line segment.

Line segments are often just called lines, for brevity. A point where two line segments meet or cross is called a vertex.

 Angles are a measure of rotation and can be measured in degrees. There are 360 degrees (written as 360◦ ) in a full turn, and therefore there are 180◦ in a half-turn and 90◦ in a quarter-turn or right angle.

Finally, a plane is a flat surface that extends infinitely far in all directions. For example, a flat piece of paper is part of a plane.

The notation and symbols used for line segments and angles are also used to refer to the lengths of line segments and to the sizes of angles.

Angles on a straight line

Since a straight angle is 180◦, any angles that together make up a straight angle add up to 180◦ . For example, in Figure 4, ∠ABC and ∠CBD together add up to 180◦. So, since ∠ABC = 30◦ , A B C D 30◦ Figure 4 ∠CBD = 180◦− 30◦ = 150◦.

The general result is summarised below.

 Angles on a straight line add up to 180◦.

Many of the activities in this unit, and many of the applications of geometry, involve using a geometric diagram to deduce the sizes of angles or the lengths of line segments.

Example 1 Calculating angles

 (a) Calculate ∠ABD in the diagram below.

 (b) Calculate the angle θ in the diagram below.

Solution

(a)       State the facts that you are going to use.

 ABC is a straight line, and angles on a straight line add up to 180◦ . Write down an equation involving the unknown angle, and solve it. So 45◦+ 80◦+ ∠ABD = 180◦ ∠ABD = 180◦− 80◦− 45◦ ∠ABD = 55◦.

(b)       Angles in a full turn add up to 360◦.

So θ + θ + θ + 60◦ = 360◦

 3θ + 60◦ = 360◦

3θ = 300◦

θ = 100◦.