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....

5 years later... I wonder if Sal is still working on it.

What happens to the measure of the inscribed angle when its vertex is on the arc? Will it be covered in the future lecture?

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).

can I use ψ as a variable to measure any angle I want to?

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.

is it possible to prove case c without proving a & b first?

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.

What is the greatest measure possible of an inscribed angle of a circle?

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

Do all questions have the lines colored? If not, how would you distinguish between the two?

Normally, to distinguish between two lines, you would have letters instead. E.g: f(x) vs g(x)

Why do you write m in front of the angle sign?

m=measure so it would just be the measure of the angle

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.

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.

Main content

Course: Integrated math 2 > Unit 11

Lesson 7: Inscribed angles

Inscribed angle theorem proof

Q: Do all questions have the lines colored? If not, how would you distinguish between the two?

Normally, to distinguish between two lines, you would have letters instead. E.g: f(x) vs g(x)

Q: Why do you write m in front of the angle sign?

m=measure so it would just be the measure of the angle

Q: 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.

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.

Google Classroom

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:

Using the image, match the variables to the terms.

1

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

(ψ)

and a central angle

(θ)

intercept the same arc: The measure of the central angle is double the measure of the inscribed angle.

θ = 2 ψ

Proof overview

To prove

θ = 2 ψ

for all

θ

and

ψ

(as we defined them above), we must consider three separate cases:

Case A	Case B	Case C
Case A has three points on the circle. The second point is less than ninety degrees clockwise from the first point. The third point is one hundred eighty degrees clockwise from the first point. The angle made by the first point, the center, and the second point is labeled theta. The angle made by the first point, the third point which passes through the center, and the second point is labeled psi.	In Case B, there are three points on the circle. The second point is more than ninety degrees clockwise from the first point. The third point is more than one hundred eighty degrees clockwise from the first point. The angle made by the first point, the center, and the second point is labeled theta. The angle made by the first point, the third point, and the second point is labeled psi.	In Case C there are three points on the circle. The second point is less than ninety degrees clockwise from the first point. The third point is less than one hundred eighty degrees clockwise from the first point. The angle made from the first point, the center, and the second point is labeled theta. The angle made from the first point, the third point, and the second point is labeled psi.

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, $ψ$ ‍.

Step 1: Spot the isosceles triangle.

Segments

\overset{―}{B C}

and

\overset{―}{B D}

are both radii, so they have the same length. This means that

△ C B D

is isosceles, which also means that its base angles are congruent:

m ∠ C = m ∠ D = ψ

Step 2: Spot the straight angle.

Angle

∠ A B C

is a straight angle, so

\begin{aligned} θ + m ∠ D B C & = 180^{\circ} \\ m ∠ D B C & = 180^{\circ} - θ \end{aligned}

Step 3: Write an equation and solve for $ψ$ ‍.

The interior angles of

△ C B D

are

ψ

ψ

, and

(180^{\circ} - θ)

, and we know that the interior angles of any triangle sum to

180^{\circ}

\begin{aligned} ψ + ψ + (180^{\circ} - θ) & = 180^{\circ} \\ 2 ψ + 180^{\circ} - θ & = 180^{\circ} \\ 2 ψ - θ & = 0 \\ 2 ψ & = θ \end{aligned}

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, $ψ$ ‍.

Step 1: Get clever and draw the diameter

Using the diameter, let's break

ψ

into

ψ_{1}

and

ψ_{2}

and

θ

into

θ_{1}

and

θ_{2}

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

(1) θ_{1} = 2 ψ_{1}

and

(2) θ_{2} = 2 ψ_{2}

because of what we learned in Case A.

Step 3: Add the equations.

\begin{aligned} θ_{1} + θ_{2} & = 2 ψ_{1} + 2 ψ_{2} & Add (1) and (2) \\ (θ_{1} + θ_{2}) & = 2 (ψ_{1} + ψ_{2}) & Group variables \\ θ & = 2 ψ & θ = θ_{1} + θ_{2} and ψ = ψ_{1} + ψ_{2} \end{aligned}

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:

θ_{2}

and

ψ_{2}

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:

(1) θ_{2} = 2 ψ_{2}

(2) (θ_{2} + θ) = 2 (ψ_{2} + ψ)

Step 3: Substitute and simplify.

\begin{aligned} (θ_{2} + θ) & = 2 (ψ_{2} + ψ) & (2) \\ (2 ψ_{2} + θ) & = 2 (ψ_{2} + ψ) & θ_{2} = 2 ψ_{2} \\ 2 ψ_{2} + θ & = 2 ψ_{2} + 2 ψ \\ θ & = 2 ψ \end{aligned}

And we're done! We proved that

θ = 2 ψ

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
Case A has three points on the circle. The second point is less than ninety degrees clockwise from the first point. The third point is one hundred eighty degrees clockwise from the first point. The angle made by the first point, the center, and the second point is labeled theta. The angle made by the first point, the third point which passes through the center, and the second point is labeled psi.	In Case B, there are three points on the circle. The second point is more than ninety degrees clockwise from the first point. The third point is more than one hundred eighty degrees clockwise from the first point. The angle made by the first point, the center, and the second point is labeled theta. The angle made by the first point, the third point, and the second point is labeled psi.	In Case C there are three points on the circle. The second point is less than ninety degrees clockwise from the first point. The third point is less than one hundred eighty degrees clockwise from the first point. The angle made from the first point, the center, and the second point is labeled theta. The angle made from the first point, the third point, and the second point is labeled psi.

In Case A, we spotted an isosceles triangle and a straight angle. From this, we set up some equations using

ψ

and

θ

. With a little algebra, we proved that

θ = 2 ψ

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

Case B	Case C
In Case B, there are three points on the circle. The second point is more than ninety degrees clockwise from the first point. The third point is more than one hundred eighty degrees clockwise from the first point. The angle made by the first point, the center, and the second point is labeled theta. The angle made by the first point, the third point, and the second point is labeled psi. A point is also on the circle to make a line segment that passes through the center to the third point as a diameter. Angle theta one is on the left and theta two is on the right of the diameter where theta was located. Angle psi one is on the left and angle psi two is on the right of the diameter located where psi was.	There are three points on the circle. The second point is less than ninety degrees clockwise from the first point. The third point is less than one hundred eighty degrees clockwise from the first point. The angle made from the first point, the center, and the second point is labeled theta. The angle made from the first point, the third point, and the second point is labeled psi. A point is on the circle with a line segment connecting it though the center to the third point making a diameter. The angle from the new point to the center to the first point is labeled theta two. The angle made by the center point, the third point, and the first point is labeled psi two.

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

θ = 2 ψ

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Pranav
Posted 5 years ago. Direct link to Pranav's post “I need help in the proofs...”
I need help in the proofs for Case 3 in inscribed angles
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toma.gevorkyan8
Posted 7 years ago. Direct link to toma.gevorkyan8's post “Hi Sal, I have a question...”
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....
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- gavinjanz24
  Posted 2 years ago. Direct link to gavinjanz24's post “5 years later... I wonder...”
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kjohnson8937
Posted a year ago. Direct link to kjohnson8937's post “can I use ψ as a variable...”
can I use ψ as a variable to measure any angle I want to?
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(4 votes)
Answer
- kubleeka
  Posted a year ago. Direct link to kubleeka's post “Yes, and it doesn't have ...”
  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.
  Comment on kubleeka's post “Yes, and it doesn't have ...”
  (7 votes)
Akira
Posted 3 years ago. Direct link to Akira's post “What happens to the measu...”
What happens to the measure of the inscribed angle when its vertex is on the arc? Will it be covered in the future lecture?
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Answer
- Reynard Seow
  Posted 2 years ago. Direct link to Reynard Seow's post “If the vertex of the insc...”
  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).
  Button navigates to signup page
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pandabuff2016
Posted 9 months ago. Direct link to pandabuff2016's post “is it possible to prove c...”
is it possible to prove case c without proving a & b first?
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- jonhlhn.surf
  Posted 8 months ago. Direct link to jonhlhn.surf's post “You do not need to prove ...”
  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.
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Jason Showalter
Posted 4 years ago. Direct link to Jason Showalter's post “What is the greatest meas...”
What is the greatest measure possible of an inscribed angle of a circle?
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Answer
- Pat Florence
  Posted 4 years ago. Direct link to Pat Florence's post “If the angle were 180, th...”
  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
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Do all questions have the lines colored? If not, how would you distinguish between the two?
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- victoriamathew12345
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  Normally, to distinguish between two lines, you would have letters instead.
  E.g: f(x) vs g(x)
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Why do you write m in front of the angle sign?
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- KC
  Posted 4 years ago. Direct link to KC's post “m=measure so it would jus...”
  m=measure so it would just be the measure of the angle
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how can i solve this
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Trinity Kelly
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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.
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- kubleeka
  Posted 5 years ago. Direct link to kubleeka's post “When you compute C/2π, be...”
  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.
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Integrated math 2

Course: Integrated math 2 > Unit 11

Getting started

What we're about to prove

Proof overview

Case A: The diameter lies along one ray of the inscribed angle, ψ‍ .

Step 1: Spot the isosceles triangle.

Step 2: Spot the straight angle.

Step 3: Write an equation and solve for ψ‍ .

Case B: The diameter is between the rays of the inscribed angle, ψ‍ .

Step 1: Get clever and draw the diameter

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

Step 3: Add the equations.

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

Step 1: Get clever and draw the diameter

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

Step 3: Substitute and simplify.

A summary of what we did

Want to join the conversation?

Case A: The diameter lies along one ray of the inscribed angle, $ψ$ ‍.

Step 3: Write an equation and solve for $ψ$ ‍.

Case B: The diameter is between the rays of the inscribed angle, $ψ$ ‍.