Inscribed Angles

How about an example?

Let's take 3 points A, B, and C lying on a circle. We'll name the center of the circle Q.

Let's also draw 2 chords, one from A to B and a second from B to C. Now we have 2 chords of a circle which share a common endpoint. So ∠ABC is an inscribed angle.

There are an infinite number of other points on the circle. We named one of these other points D. If we started to name all of them we'd soon run out of letters. You can use any 3 points on the circle to define an inscribed angle. You can have ∠ADC and ∠BCA and ∠CBD for example.

Remember how central angles have the same measure as the arc of the circle? For inscribed angles, the measure of the arc is 2 times the measure of the angle.

Can you prove that?

Sure. Remember that geometry is all about triangles and lengths. (The lengths are represented by circles; our circle is everything a certain length from Q.) So to find a proof we will start by looking for triangles with known lengths.

Draw the radius QA and then QB and QC. This gives us 2 triangles: ▵QAB and ▵QBC. Because every radius has the same length, these are isosceles triangles.

What do we know about isosceles triangles? The legs have equal length and the base angles have equal measure. In other words, ∠QAB ≅ ∠QBA and ∠QCB ≅ ∠QBC.

OK, sure, but how does that help us?

Let's label the measures of all the angles with lowercase letters. (That's just to make it easier to write down what we know.) We'll say the measure of ∠QAB is 'a' and the measure of ∠QCB is 'c'. We can also say m∠AQB = r and m∠CQB = s. Remember that the pairs of base angles have equal measures, so that's all the letters we need to use.

What does this tell us about the inscribed angle? The measure of ∠ABC is the sum of a and c:
m∠ABC = a + c

We also know something about the sum of the angles of each triangle: They add up to 180°. That is, a + a + r = 180 and c + c + s = 180.

We can add the equations together. (Well, actually we'll add 180 to each side of the first equation. But c + c + s is the same as 180. So we can write 180 in different ways.)
a + a + r + c + c + s = 180 + 180
and, simplifying, we can write
2a + 2c + r + s = 360°

But you're supposed to relate ∠ABC to the arc of the circle!

Exactly! So in addition to information about the angles, we also need information about the arcs. Fortunately, the central angle theorem gives us a connection between them. We don't really know how big r and s are, but we do know that the arcs have the same measures as the central angles.

In other words, the arc from A to B has the measure r and the arc from B to C has the measure s. The measure of the combined arc ABC is r + s. The whole circle, of course, measures 360°. So the measure of the remaining part — the rest of the circle without arc ABC — is:
m⌒ADC = 360° - (r + s)

That's important because this rest of the circle is the arc we are actually interested in. It's the arc cut off by our angle, ∠ABC.

I noticed that you highlighted 3 of the equations.

That's because I think they are the most useful. Let's work backwards. (I almost always suggest working backwards.) The last equation is about the measure of the arc which we are interested in. The logical thing is to substitute the value of 360° in the middle equation into this last equation and see what happens.

Copy the 3rd equation from above	m⌒ADC = 360° - (r + s)
Substitute the value of 360° from the middle equation	m⌒ADC = 2a + 2c + r + s - (r + s)
Use the distributive property	m⌒ADC = 2a + 2c + r + s - r - s
Simplify by combining terms	m⌒ADC = 2a + 2c
Use the distributive property the other way	m⌒ADC = 2(a + c)
Substitute the value of a + c from the first equation	m⌒ADC = 2(m∠ABC)
Which is what you asked me to prove, that	the measure of the arc is 2 times the measure of the inscribed angle.

Hey, cool. Got any corollaries?

As a matter of fact, I do. Let's suppose we slide the vertex point B along the circle. What happens to the size of the angle? It stays the same.

How can I know that? Look back at the last equation in the proof above.
m⌒ADC = 2(m∠ABC)
If the points A and C don't change, then the measure of arc ADC won't change, either. if m⌒ADC is the same, then the equation tells us that m∠ABC also remains the same.

The point B, the measures of arcs AB and BC, and the real sizes of r and s play no part in the equation. So B can move anywhere between points A and C and the angle will stay the same. Once you've picked points A and C you have defined the measure of the inscribed angle, even without knowing point B.

Of course, this also means that if you move C farther from A (or closer to A), then the measure of the inscribed angle always changes.