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Definitions:

Radius – the radius of the circle is any distance from the center of the circle, to the edge of the circle.

Basically the radius starts at the center, and moves out to the edge, like:

Diameter – the diameter of a circle is the distance from one edge of the circle, to the other edge of the circle

Chord – a chord is a segment whose endpoints lie on a circle.

Inscribed Angles – an inscribed angle is an angle whose vertex lies on a circle, and whose sides contain chords of the circle.

Central Angles – a central angle is an angle with a measure of less than or equal to 180 degrees, whose vertex lies at the center of a circle

Minor Arc – an arc whose points are on or in the interior of a corresponding central angle.

Major Arc – an arc whose points are on or in the exterior of a corresponding central angle.

Semicircle – a semicircle is an arc whose endpoints are the endpoints of a diameter.

Adjacent Arcs – adjacent arcs are arcs of the same circle that intersect in exactly one point.

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THE INSCRIBED ANGLE THEOREM

So, the inscribed angle theorem says:

“The measure of an inscribed angle is equal to half of the measure of its intercepted arc.”

Or in other words:

But, how do we know this works?

Well….

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FANTASTIC! NOW WHAT?

Well now we can actually solve problems

Because now we know that the inscribed angle is equal to ½ the alternating central angle.

Here’s an example:

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Example 1:

Solve for x:

So, we know, using the inscribed circles theorem, that the inscribed angle is half of the arc.

Since we can see the arc is 60 degrees

That means:

Since we know that the angle is also equal to 2x + 10, then:

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Inscribed Angle of Diameter Theorem

So the inscribed angle of diameter theorem says:

“The endpoints of a diameter lie on an inscribed angle if and only if the inscribed angle is a right angle.”

Basically, this is all pretty easy to understand.

To help, let’s look at a circle:

Now, the theorem says the end points of a diameter lie on an inscribed angle.

So:

Next, it says that the angle needs to be a right angle

But that makes sense right?

Because we know the diameter is equal to 180 degrees

And the inscribed angle is always half of the arc.

And since the arc is 180 degrees, then that would mean the inscribed angle is one half of 180

Or again, 90 degrees.

So, now let’s do a few more examples to make sure you understand.

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EVEN MORE DEFINITIONS:

Tangent – a tangent is a line in the same plane as a circle, that intersects the circle in only one point.

Point of Tangency – The point where a tangent intersects with a circle is called the point of tangency.

Circumscribed Angle – a circumscribed angle is an angle formed by two rays from a common endpoint that are tangent to a circle.

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Okay, now we have enough to work with

Now that we’ve finished with the definitions, we can look at the theorem that we need.

So the first theorem we want to start with is the Tangent-Radius Theorem, which says:

“If a line is tangent to a circle, then it is perpendicular to a radius drawn to the point of tangency.”

Yeah, in plain English, it means if you have a line that is tangent to a circle:

That tangent line is also going to be perpendicular to a radius of that circle:

This is true because of how tangent lines work.

Now, to be honest, the proof is a little complex, so it will be included.

This also means if you have an outside line, that touches the circle at one point and is perpendicular to a radius in the circle, then that line must be a tangent line.

This is what we call the converse.

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The Circumscribed Angle Theorem

The next theorem we need to go over is the Circumscribed Angle Theorem, which says:

“A Circumscribed angle of a circle and its associated central angle are supplementary.”

Again, in plain English it means this.

If we have a circle:

And this circle has two tangent lines that intersect (to make an circumscribed angle:

And, we have two radii that are connected to those tangent lines (like so):

Then the inside angle + the circumscribed angle = 180

Or in other words:

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Proving the Circumscribed Angle Theorem

So, to start off with, we need our circle, tangents, and radii.

Now, if we look at this shape that it’s making, it would seem that we have a quadrilateral, right?

So this means:

But, we know that angles B and D are equal to 90 degrees because of the Tangent Radius Theorem, right?

So this means that:

So, again, the circumscribed angle and the inscribed angle are supplementary.

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Example 1:

Solve for x:

Since we know we can make this into a quadrilateral, let’s do that first.

So:

Since we know that the tangent angles are 90 degrees:

Then we know their opposite sides must be equal:

So, then we can set our measurements to each other and get:

So then x = 3

Which means are tangents are:

12 * 3 = 36 and

4 * (3)^2 = 4 * 9 = 36

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Example 2:

Solve for x:

Since we know we can make this into a quadrilateral, let’s do that first.

So:

Since we know that the inscribed angle and the circumscribed angle are supplementary

Then we know they add to 180 degrees.

So to solve for this, we just:

And plugging this back into the equation, we get:

So our inscribed angle is 144 degrees.

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SO THEN, WHAT’S A QUADRILATERAL?

Well, if we recall, a quadrilateral is any shape with 4 sides.

So, why do we care?

Well because we can basically inscribe any shape we want into a circle

And the shape we’re going to deal with now is quadrilaterals.

So, to start, let’s talk about the Inscribed Quadrilateral Theorem:

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THE INSCRIBED QUADRILATERAL THEOREM

So the inscribed quadrilateral theorem says:

“If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.”

So, what does this mean?

Well, again, it means that the opposite angles of the quadrilateral will add up to 180 degrees.

So, if we have something like:

Then we know that:

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PROVING THE INSCRIBED QUADRILATERAL THEOREM

So to prove the inscribed quadrilateral theorem, first we need a circle:

And now, we need a quadrilateral inscribed in the circle:

Now, looking at this circle, we can see that:

And the inscribed angles are equal to half of their corresponding arcs.

Then we also know that:

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AWESOME, SO WHAT NOW?

Well, again, just like with the other types of angles we’ve dealt with

Now we can solve problems when given very little information.

For example:

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Example 1

Solve for x:

Since we know that the opposite angles of this quadrilateral are supplementary.

Then we know that:

So:

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Example 2

Solve for x:

Since we know that the opposite angles of this quadrilateral are supplementary.

Then we know that:

So:

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Example 3

Solve for x:

Since we know that the opposite angles of this quadrilateral are supplementary.

Then we know that:

So:

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Example 4

Solve for x:

Since we know that the opposite angles of this quadrilateral are supplementary.

Then we know that:

So: