Inscribed angle in a circle

We are here to define for you what an inscribed angle in a circle is. Also, to give you tips to remember its definition and characteristics in the most logical way.
Before talking about the inscribed angle in the circle, let's take a moment to look at its name - inscribed angle.
Its name, lets us understand that it has a connection with the circumference and indeed it does.

Now, we can move on to the definition of an inscribed angle and it will stay in our minds thanks to logic.
What is an inscribed angle in a circle?
An inscribed angle in a circle is an angle whose vertex is on the top of the circle: on the circumference of the circle and its ends are chords in the circle.
Let's see it in the figure:

We have a circle in front of us.
We mention that an inscribed angle is an angle whose vertex is on the circle, that is, on the circumference.
and whose ends are chords in a circle.
Therefore, if you draw any two chords in a circle, they will meet at the same point on the circumference - On the circle itself, we will create an angle.
The angle that will be formed, will be an inscribed angle in the circle.
We will mark A at some point on the circle and two chords inside the circle that meet at point $A$.

Now that we know what an inscribed angle in a circle is and can easily identify it,
we must know some important theorems and properties of an inscribed angle in a circle.
Shall we start?
Equal inscribed angles
When can we determine that the inscribed angles in a circle are equal?

That is, if there are any chords on which inscribed angles from the same side lean, they will be equal.

Let's see this in the figure:

Here is a circle and a chord $AB$.
We can see that angles 1,2,3
lean on the chord AB from the same side and therefore are equal.

Example of angles leaning on the same chord but not from the same side:

We can see that angles 1 and 2 do lean on the same chord but not from the same side and therefore we cannot determine that they are equal.

That is, if we are given that there are equal inscribed angles, we can determine that the chords and the arcs they intercept are also equal.
Let's see this in the figure:

Before us is a circle
if we are given that:
$∢1=∢2$
Then we can determine that:
$AB = DC$
and also
$AB = DC$

That is, if we are given equal arcs in a circle, we can determine that the inscribed angles opposite them are equal.
Let's see this in the figure:

Before us is a circle.
If we are given that:
$AB = CD$
then
$∢1=∢2$

Now we will study the relationship between an inscribed angle in a circle and a central angle in a circle.
Remember that a central angle in a circle is an angle whose vertex is at the center of the circle and whose ends are radii in the circle.
Like here:

In a circle, the inscribed angle will be half of the central angle that leans on the same arc.

That is:
If in the circle we identify a central angle and an inscribed angle that lean on the same arc, we can say that as they lean on the same arc, the inscribed angle will be equal to half of the central angle.
Let's see this in the figure:

Remember, the diameter is the largest radius in a circle: a line that connects 2 points on the top of the circle and passes through the center of the circle.
What is the relationship between this and the circumferential angle? Excellent that you asked.
An inscribed angle that leans on a diameter equals $90°$ degrees.
In the same way, we can say that if any inscribed angle in a circle equals $90°$ degrees, it leans on a diameter.
Let's see this in the figure:

If the diameter $AB$
then
$∢ACB = 90°$
in the same way, if
$∢ACB=90°$
then
$AB$ is the diameter.

Do you remember we talked about the fact that inscribed angles leaning on the same chord on one side are equal?
Now, we are talking about inscribed angles leaning on the same chord but on its two different sides. The two angles together add up to $180°$ degrees.
Let's see this in the figure:

In front of us, there is a circle and a chord $AB$
The angles $\sphericalangle ACD$ and $\sphericalangle ADB$
are inscribed angles that lean on the same chord on its two different sides and therefore their sum will be $180°$.

If you are interested in this article, you might also be interested in the following articles:

• The center of the circle
• Circle
• Radius
• Diameter
• Pi
• The circumference perimeter
• Circular area
• Arcs in a circle
• Chords in a circle
• Central angle in a circle
• Perpendicular to the chord from the center of the circle
• Distance of the chord to the center of the circle

In the Tutorela blog, you will find a variety of articles about mathematics.