Lesson 28: Angles

New Concept

An angle is formed where lines or rays intersect. Angles are classified as acute (less than a right angle), right (a square angle), or obtuse (greater than a right angle).

What’s next

Next, you’ll apply these definitions in worked examples, learning to identify and correctly name acute, right, and obtuse angles in various figures.

Property

When two lines are drawn in the same plane, they will either cross at one point or they will not cross at all. When lines do not cross but stay the same distance apart, we say that the lines are parallel. When lines cross, we say that they intersect. When they intersect and make square angles, we call the lines perpendicular.

Examples

Railway tracks are a real-world example of parallel lines.
The letter X is formed by two lines that intersect.
The corners of a square window pane are formed by perpendicular lines.

Explanation

Think of a flat tabletop as your plane. Lines are like endless pencil strokes. They can be best buddies, running side-by-side forever (parallel), or they can cross paths (intersect). If they cross to form a perfect 'T' shape, they're perpendicular, like the crossing streets on a map. Otherwise, they are just oblique.

Property

Where lines intersect, angles are formed. Rays make up the sides of the angles. The rays of an angle originate at a point called the vertex of the angle.

Examples

The hands of a clock form an angle with the vertex at the center.
A slice of pizza has an angle at its pointy tip.
The corner of a book page is an angle where two edges meet.

Explanation

Imagine two laser beams shooting out from the same point! That point is the 'vertex,' and the beams are the 'rays.' The space or opening between those two laser beams is the angle. It's all about how much 'turn' there is between the two sides, starting from their shared corner.

Property

Naming an angle correctly is like giving exact map directions. There are three common ways to name an angle:

1. By its vertex: Use only the letter of the corner point (e.g., ∠B).
2. By three points: Use a point on one side, the vertex, and a point on the other side. The vertex MUST be in the middle (e.g., ∠ABC or ∠CBA).
3. By a number: Use a single number placed inside the angle near the vertex (e.g., ∠1).

Examples

• A simple angle with corner X can be called ∠X.
• The same angle with points Y and Z on its straight rays can be named ∠YXZ or ∠ZXY. (Never ∠XYZ, because the vertex X must be in the middle!)
• In a crowded diagram with many lines crossing, an angle might just have a "3" written inside it. We call this ∠3.

Explanation

If an angle is standing all by itself, just calling it by its corner name (like ∠B) is perfectly fine. But when many angles share the same corner, calling them all "∠B" would be confusing! That is when you must use the 3-letter method to trace the exact path, or use the quick number method to easily identify the specific angle you are talking about.