Certain shapes, and configurations of lines which make them up have particular angular properties which can allow you to determine missing angles connected to them. These properties are important to know, as they can help you solve trigonometry problems by helping you identify needed angles based on other angles, even if you don’t know the angles in the triangles.
Straight Line And Connecting Line
Where you have a straight line and a line which connects to it, the sum of the angles made on the side of the connecting line will equal 180°.
$$a+b=180$$
This is because a straight line has an angle of 180° and thus the total of the angles around the connecting point must be one angle of 180° (which is opposite the connecting line), and a number of angles around the point where these two lines meet which add up to 180° (Figure 1).
This property is the basis for the other angle properties described here, and is also a useful property in its own right for solving some trigonometry problems. Problems involving ambiguous triangles for example involve a straight line of ambiguous length, with lines of known length connecting to that line (Figure 2).
Intersecting Lines
When a line doesn’t just meet another line, but also crosses it, there are two sets of lines connecting to a straight line around the same point. As these lines are straight, there will be two pairs of identical angles, which add up to 360°.
$$2a+2b=360$$
The orientation of these angles around the intersecting point are that angles opposite each other are identical (Figure 3). This is because since the first rule is that the angles created by a line and another line connecting to it will add up to 180°, as the lines are straight, any one of the angles, and a single adjacent angle (which will be an angle a and angle b) will add up to 180°.
This rule can be used to solve trigonometry problems where two triangles meet each other at a corner. This is because these often contain intersecting lines, and so angles can be worked out by using this rule about intersecting lines (Figure 4).
Intersecting With Parallel Lines
When a line intersects with a pair of parallel lines, it creates two intersection points. These intersection points both have the properties described above, and are identical to one another, but transposed to a different location on the line which intersects both of the parallel lines (Figure 5). This means that the angles from one of these intersections can be transposed into the other.
A use of this rule is that it shows that angles of elevation and depression are identical when looking at problems involving right triangles (Figure 6). This is because the angle of elevation is looking up from the baseline, and the angle of depression is looking down from a line parallel to this baseline, thus this is a case where a line intersects two parallel lines.
