Subject: Geometry

# Angles and Intersecting Lines

The Breakdown
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## 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?**

They do not share the same vertex or line segment (even though they look like they'd be able to fit together).

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?

**Answers:**

1. They are equal in measure

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