Foundations — Types of angles — acute, right, obtuse, straight, reflex, complete

Before you can classify angles, you must know exactly what each ingredient means — with no cheating, no "you already know this." This page builds every symbol the parent note leans on, from the ground up. We go in an order where each idea rests on the one before it.

1. The point and the ray — the raw material

Before an angle exists, we need two ingredients.

Look at the figure. The black dot is the starting point. The red arrow is the ray — the little dot marks where it begins, and the arrowhead reminds you it keeps going forever. A ray points in exactly one direction.

Why the topic needs this: an angle is built from two rays. Without the idea of "a direction starting from a point," there is nothing to measure the turn between.

2. The vertex and the angle itself

Now bring two rays together so they share the same starting point.

In the figure, both rays start at the same black dot (the vertex). The red shaded wedge is the angle — it is not the rays and not the corner, it is the spread between them. A wide wedge means a big turn; a narrow wedge means a small turn.

Why the topic needs this: every type — acute, obtuse, reflex — is a statement about how big this turn is. So we need a way to put a number on the turn. That number is the degree.

3. The degree — the unit that puts a number on a turn

To compare turns, we need a measuring unit, just like we need centimetres to compare lengths.

So the symbol $°$ (a raised little circle) means "degrees." When we write $47°$, we are saying "47 of those 360 slices of a full turn."

The figure shows one full turn sliced into equal degree-marks (drawn coarsely so you can see them). Reading around from the red starting ray:

• a quarter of the way round is $\frac{360}{4}=90°$,
• halfway round is $\frac{360}{2}=180°$,
• all the way round back to start is $360°$.

Why the topic needs this: every classification is a comparison of numbers of degrees — "is this turn less than 90 degrees?" You cannot ask that question until "degree" exists.

4. The symbol $\theta$ — a name for an unknown angle

The parent note writes things like $0°<\theta <90°$. What is $\theta$?

We use a Greek letter purely by tradition, so that reading a formula you instantly think "ah, that stands for an angle." Other common angle-letters are $\alpha$ (alpha) and $\beta$ (beta), which the parent uses for two angles at once.

Why the topic needs this: to state a rule that works for every angle at once ("any angle below 90° is acute"), we need one symbol that stands for whatever angle you hand it. That is $\theta$.

5. The comparison symbols $<$, $>$ and "between"

The parent note classifies angles with lines like $90°<\theta <180°$. Every symbol there must be earned.

So $47°<90°$ reads "47 degrees is smaller than 90 degrees."

Why the topic needs this: the six types are defined by ranges of degrees, and $<$/$>$ are exactly the tools that carve the number line into those ranges.

6. Landmark turns — the fenceposts between the types

Four special turn-sizes act as fenceposts that separate the six categories. They come straight from slicing the full turn:

These are not new ideas — each is just "some fraction of the 360-slice full turn." They matter because the names switch as you cross each fencepost:

Why the topic needs this: classification is literally "which two fenceposts does $\theta$ fall between?"

7. The two angles at one corner — where "reflex" comes from

Here is the subtle idea the parent note relies on for reflex angles. When two rays share a vertex, they actually cut the full turn into two pieces, not one.

That single equation is why the parent's reflex formula works: ${\theta }_r=360°-\theta$ It is not a new law — it is just "the leftover of the full turn after you take the short way."

Why the topic needs this: without seeing that a corner holds two angles, "reflex" looks like nonsense ("how can an angle be bigger than a straight line?"). Once you see the long-way sweep, reflex angles are obvious.

Prerequisite map

Each foundation on the left must be solid before "Types of angles" on the right makes sense. Follow the arrows back to the parent: Types of angles — acute, right, obtuse, straight, reflex, complete.

Where these foundations go next

• To actually read a degree number off a drawing you use a protractor.
• Splitting one angle into two equal halves is done with an angle bisector.
• The $90°$ and $180°$ landmarks power Complementary and supplementary angles.
• The "two angles at a corner" idea grows into Adjacent angles and linear pairs and Vertically opposite angles.

Equipment checklist

Test yourself — cover the right side and answer before revealing.