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° (Eq. 1).

Eq. 1: \(a+b=180\)

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).

Figure 1: Two angles that intersect along a straight line will add up to 180°. Thus, if you know one of the angles, you can find the second angle using algebra and rearranging the equation to find the missing angle.

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).

Figure 2: The property of lines meeting a straight line creating angles which add up to 180° can be used to solve ambiguous triangles. In ambiguous triangles, there are two places where lines meet a straight line. This, combined with the fact that there is an isosceles triangle, allows the possible triangles to be identified in the ambiguous triangles..

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° (Eq. 2).

Eq. 2: \(2a+2b=360\)

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°.

Figure 3: When two lines cross each other, the angles opposite one another are identical, and all angles add up to 360°. This can be used to solve all of the angles in the triangle, if one angle is known.

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).

Figure 4: If two triangles are adjacent to one another and connected via one of its corners with two crossing straight lines, the angles of all sides of the triangle can be determined using the properties of angles around intersecting lines.

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.

Figure 5: When parallel lines intersect one another, the angles of all intersection points, and their arrangements are identical to one another. This means that angles along lines which are parallel to the first angle can be determined indefinitely, provided that the lines are parallel.

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.

Figure 6: The property of intersecting parallel lines means that the angle of elevation (an observer looking up at an object) and depression (an observer looking down at an object) are identical to one another relative to one another.