Angles and Intersecting Lines

The basics of angles

What is an angle?
The word ‘angle’ derives from the Latin word angulus, meaning ‘corner’. An angle is a measurement of the amount of rotation required around a vertex to make two intersecting lines lie upon each other.

Because that might be a bit of a mouthful, a simpler definition is that it’s a measure of how far apart the direction two intersecting lines take. That’s not a perfect dictionary definition, but it’s OK for the basics. If you’re a perfectionist, read the first one a few times slowly until it makes sense.

Any number of angles around a point will always add to exactly 360°.

Figure 1: Any number of angles around a point will add up to 360 degrees

This is simply one of the fundamental laws of geometry, in the same sense that within one kilometre is exactly one thousand metres. No matter how many lines intersect a point, if we go all the way around adding up all the angles, they will always total 360°.

Test this theory by adding the angles within these figures:

Figure 2: Angles adding up to 360 degrees

Figure 3: Angles adding up to 360 degrees

Interior and Exterior Angles

When discussing angles made by intersecting lines we’ll sometimes mention interior and exterior. These words can refer to either sides or angles. Have a look at the following diagram.

Figure 4: Showing Interior vs Exterior Angles

Pretty simple, huh?

Adjacent Angles

Adjacent angles share a vertex and a line segment.

Figure 5: An Adjacent Angle

In this picture, both of these angles x and y share the vertex at point B, and the line segment they both share is BD.

Figure 6: Multiple Adjacent Angles

In this example \angle A is adjacent to \angle B and \angle H , but not to angles C, D, E, F and G. \angle E is adjacent to \angle D and \angle F but not to any of the others.

Please consider that adjacent angles will always be congruent if their exterior sides are perpendicular.

Question 1:

These angles are not adjacent…. Why not?

Figure 7: Why are these angles not adjacent?

However, they are supplementary.

Supplementary Angles

Two angles are supplementary if their measures total 180°. Each angle is said to be the supplement of the other.

Supplementary Non-Adjacent Angles

Figure 8: Supplementary Angles

Supplementary Adjacent Angles

Figure 9: Supplementary Adjacent Angles

Two adjacent angles will always be supplementary if they lie along a straight line.

Complimentary Angles

Two angles are complementary if their measures total 90°.

Complimentary Non-Adjacent Angles

Figure 10: Complimentary Non-Adjacent Angles

Complimentary Adjacent Angles

Figure 11: Complimentary Adjacent Angles

(If we weren’t using a box to show the two angles add to 90°, we’d write \angle A + \angle B = 90°.)

Congruent Angles

These are non-adjacent angles made by intersecting lines. In the following diagram, we can see \angle a and \angle c are congruent. Angles b and d are also congruent.

Figure 12: Congruent Angles

Question 2:

From this example, what property do vertical lines possess?