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## High school geometry

# Inscribed angle theorem proof

Proving that an inscribed angle is half of a central angle that subtends the same arc.

## Getting started

Before we get to talking about the proof, let's make sure we understand a few fancy terms related to circles.

Here's a short matching activity to see if you can figure out the terms yourself:

Nice work! We'll be using these terms through the rest of the article.

## What we're about to prove

We're about to prove that something cool happens when an inscribed angle left parenthesis, start color #11accd, \psi, end color #11accd, right parenthesis and a central angle left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis intercept the same arc:

**The measure of the central angle is double the measure of the inscribed angle.**## Proof overview

To prove start color #aa87ff, theta, end color #aa87ff, equals, 2, start color #11accd, \psi, end color #11accd for all start color #aa87ff, theta, end color #aa87ff and start color #11accd, \psi, end color #11accd (as we defined them above), we must consider three separate cases:

Case A | Case B | Case C |
---|---|---|

Together, these cases account for all possible situations where an inscribed angle and a central angle intercept the same arc.

## Case A: The diameter lies along one ray of the inscribed angle, start color #11accd, \psi, end color #11accd.

#### Step 1: Spot the isosceles triangle.

Segments start overline, start color #e84d39, B, C, end color #e84d39, end overline and start overline, start color #e84d39, B, D, end color #e84d39, end overline are both radii, so they have the same length. This means that triangle, C, B, D is isosceles, which also means that its base angles are congruent:

#### Step 2: Spot the straight angle.

Angle angle, start color #e84d39, A, B, C, end color #e84d39 is a straight angle, so

#### Step 3: Write an equation and solve for start color #11accd, \psi, end color #11accd.

The interior angles of triangle, C, B, D are start color #11accd, \psi, end color #11accd, start color #11accd, \psi, end color #11accd, and left parenthesis, 180, degrees, minus, start color #aa87ff, theta, end color #aa87ff, right parenthesis, and we know that the interior angles of any triangle sum to 180, degrees.

Cool. We've completed our proof for Case A. Just two more cases left!

## Case B: The diameter is between the rays of the inscribed angle, start color #11accd, \psi, end color #11accd.

#### Step 1: Get clever and draw the diameter

Using the diameter, let's break start color #11accd, \psi, end color #11accd into start color #11accd, \psi, start subscript, 1, end subscript, end color #11accd and start color #11accd, \psi, start subscript, 2, end subscript, end color #11accd and start color #aa87ff, theta, end color #aa87ff into start color #aa87ff, theta, start subscript, 1, end subscript, end color #aa87ff and start color #aa87ff, theta, start subscript, 2, end subscript, end color #aa87ff as follows:

#### Step 2: Use what we learned from Case A to establish two equations.

In our new diagram, the diameter splits the circle into two halves. Each half has an inscribed angle with a ray on the diameter. This is the same situation as Case A, so we know that

and

because of what we learned in Case A.

#### Step 3: Add the equations.

Case B is complete. Just one case left!

## Case C: The diameter is outside the rays of the inscribed angle.

#### Step 1: Get clever and draw the diameter

Using the diameter, let's create two new angles: start color #ed5fa6, theta, start subscript, 2, end subscript, end color #ed5fa6 and start color #e07d10, \psi, start subscript, 2, end subscript, end color #e07d10 as follows:

#### Step 2: Use what we learned from Case A to establish two equations.

Similar to what we did in Case B, we've created a diagram that allows us to make use of what we learned in Case A. From this diagram, we know the following:

#### Step 3: Substitute and simplify.

And we're done! We proved that start color #aa87ff, theta, end color #aa87ff, equals, 2, start color #11accd, \psi, end color #11accd in all three cases.

## A summary of what we did

We set out to prove that the measure of a central angle is double the measure of an inscribed angle when both angles intercept the same arc.

We began the proof by establishing three cases. Together, these cases accounted for all possible situations where an inscribed angle and a central angle intercept the same arc.

Case A | Case B | Case C |
---|---|---|

In Case A, we spotted an isosceles triangle and a straight angle. From this, we set up some equations using start color #11accd, \psi, end color #11accd and start color #7854ab, theta, end color #7854ab. With a little algebra, we proved that start color #aa87ff, theta, end color #aa87ff, equals, 2, start color #11accd, \psi, end color #11accd.

In cases B and C, we cleverly introduced the diameter:

Case B | Case C |
---|---|

This made it possible to use our result from Case A, which we did. In both Case B and Case C, we wrote equations relating the variables in the figures, which was only possible because of what we'd learned in Case A. After we had our equations set up, we did some algebra to show that start color #aa87ff, theta, end color #aa87ff, equals, 2, start color #11accd, \psi, end color #11accd.

## Want to join the conversation?

- I need help in the proofs for Case 3 in inscribed angles(12 votes)
- Hi Sal, I have a question about the angle theorem proof and I am curious what happened if in all cases there was a radius and the angle defined would I be able to find the arch length by using the angle proof? Or I had to identify the type of angle that I am given to figure out my arch length? Thanks....(6 votes)
- 5 years later... I wonder if Sal is still working on it.(5 votes)

- can I use ψ as a variable to measure any angle I want to?(3 votes)
- Yes, and it doesn't have to be an angle. You can assign any variable you like to any symbol you like. You can use Latin letters, Greek letters, Hebrew letters, random shapes, emoji, or anything else.

It's common practice to use the variables θ, φ, ψ for angle measures (I myself like to use η, since it's the letter before θ), but the rules aren't set in stone. Define whatever you like.(4 votes)

- What happens to the measure of the inscribed angle when its vertex is on the arc? Will it be covered in the future lecture?(4 votes)
- If the vertex of the inscribed angle is on the arc, then it would be the reflex of the center angle that is 2 times of the inscribed angle. You can probably prove this by slicing the circle in half through the center of the circle and the vertex of the inscribed angle then use Thales' Theorem to reach case A again (kind of a modified version of case B actually).(1 vote)

- is it possible to prove case c without proving a & b first?(3 votes)
- You do not need to prove case B to prove case C, or vice-verse. But in proving case C (or proving case B), you need to prove case A first/along the way.(2 votes)

- Why do you write m in front of the angle sign?(1 vote)
- m=measure so it would just be the measure of the angle(5 votes)

- What is the greatest measure possible of an inscribed angle of a circle?(2 votes)
- If the angle were 180, then it would be a straight angle and the sides would form a tangent line. Anything smaller would make one side of the angle pass through a second point on the circle. So the restriction on the inscribed angle would be:

0 < ψ < 180(4 votes)

- Ok so I have a small question, I'm doing something called VLA and they gave me two different equations one to find the radius using the circumference, and the other to find the diameter also using the circumference, the equations were. Circumference/p = diameter, and the other was circumference/2p = radius, but i'm confused cause when I used the second one, it would give me a really big number while the first equation gave me a smaller number. Also sorry if this has nothing to do with what you were talking about Sal, I was waiting until I had enough energy to be able to ask my question.(1 vote)
- When you compute C/2π, be sure that you're dividing by π by putting the denominator in parentheses. If you just enter C/2*π, the calculator will follow order of operations, computing C/2, then multiplying the result by π.(5 votes)

- Do all questions have the lines colored? If not, how would you distinguish between the two?(2 votes)
- Normally, to distinguish between two lines, you would have letters instead.

E.g: f(x) vs g(x)(2 votes)

- I don't understand was a radian angle is and how to get the circumference from it. I also mess up when fractions and the pie symbol are used.(2 votes)
- The radians for an angle are based on how many radii equal the length of the same arc subtended by that angle. In relation to the circumference, the circumference is equal to 2(pi)(r) r meaning radius, not radians (there is a difference). The circumference can also be seen as the arc for the whole circle and in an arc there are 2 pi radii, so there are 2 pi radians in a whole entire circle.(2 votes)