Inscribed angle theorem proof (article) | Khan Academy (2024)
Proving that an inscribed angle is half of a central angle that subtends the same arc.
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Pranav
5 years agoPosted 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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(16 votes)
toma.gevorkyan8
7 years agoPosted 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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(8 votes)
gavinjanz24
2 years agoPosted 2 years ago. Direct link to gavinjanz24's post “5 years later... I wonder...”
5 years later... I wonder if Sal is still working on it.
(11 votes)
kjohnson8937
2 years agoPosted 2 years 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?
2 years agoPosted 2 years 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.
(6 votes)
Jason Showalter
4 years agoPosted 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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(4 votes)
Pat Florence
4 years agoPosted 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
(5 votes)
Akira
3 years agoPosted 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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(5 votes)
Reynard Seow
3 years agoPosted 3 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).
(2 votes)
pandabuff2016
10 months agoPosted 10 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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(4 votes)
jonhlhn.surf
10 months agoPosted 10 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.
(3 votes)
taylor k.
4 years agoPosted 4 years ago. Direct link to taylor k.'s post “Do all questions have the...”
Do all questions have the lines colored? If not, how would you distinguish between the two?
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(3 votes)
victoriamathew12345
3 years agoPosted 3 years ago. Direct link to victoriamathew12345's post “Normally, to distinguish ...”
Normally, to distinguish between two lines, you would have letters instead. E.g: f(x) vs g(x)
(3 votes)
Konstantin Zaytsev
4 years agoPosted 4 years ago. Direct link to Konstantin Zaytsev's post “Why do you write m in fro...”
Why do you write m in front of the angle sign?
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(1 vote)
KC
4 years agoPosted 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
(5 votes)
eperez3463
a year agoPosted a year ago. Direct link to eperez3463's post “how can i solve this”
how can i solve this
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(3 votes)
Trinity Kelly
5 years agoPosted 5 years ago. Direct link to Trinity Kelly's post “Ok so I have a small ques...”
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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(1 vote)
kubleeka
5 years agoPosted 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.
If we know the measure of the central angle with shared endpoints, then the inscribed angle is just half of that angle. If we know the measure of the arc our inscribed angle intercepts, we just divide that in half to get the measure of the inscribed angle.
If both the triangles are superimposed on each other, we see that ∠B =∠E and ∠C =∠F. And the non-included sides AB and DE are equal in length. Therefore, we can say that ∆ABC ≅ ∆DEF.
Proof: In the quadrilateral ABCD can be inscribed in a circle, then we have seen above using the inscribed angle theorem that the sum of either pair of opposite angles = (1/2(a1 + a2 + a3 + a4) = (1/2)360 = 180. Conversely, if the quadrilateral cannot be inscribed, this means that D is not on the circumcircle of ABC.
Because a semicircle (half a circle) creates an intercepted arc that measures 180°, therefore, any corresponding inscribed angle would measure half of it, as Varsity Tutors nicely states.
The measure of an inscribed angle is equal to half of the measure of the arc between its sides. Considering that the arc of a semicircle is 180º, any angle inscribed in a semicircle has half that value, that is 90º. Any angle inscribed in a semicircle is right.
The measure of an inscribed angle is equal to half the measure of the central angle that goes with the intercepted arc. The measure of an inscribed angle is equal to half the measure of its intercepted arc.
Corollary (Inscribed Angles Conjecture III ): Any angle inscribed in a semi-circle is a right angle. Proof: The intercepted arc for an angle inscribed in a semi-circle is 180 degrees. Therefore the measure of the angle must be half of 180, or 90 degrees. In other words, the angle is a right angle.
You can draw an equilateral triangle inside the circle, with vertices where the circle touches the outer triangle. Now, you know how to calculate the area of that inner triangle from Sal's video. Specifically, this is 3/4 * r^2 * sqrt(3). (When r=2 like in the video, this is 3 * sqrt(3).)
Inscribed Angle Property. Inscribed angles subtended by the same arc are congruent (equal in measure). CBD are congruent (equal in measure), since both are inscribed angles subtended by arc(CD).
The vertex of an inscribed angle can be anywhere on the circle as long as its sides intersect the circle to form an intercepted arc. The Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. Inscribed angles that intercept the same arc are congruent.
An inscribed triangle is a triangle inside a circle. To draw an inscribed triangle, you first draw your triangle. Then you draw perpendicular bisectors for each side of the triangle.
In order to use AAS, all that is necessary is identifying two equal angles in a triangle, then finding a third side adjacent to only one of the angles in each of the triangles such that the two sides are equal. This is enough to prove the two triangles are congruent.
Imagine translating one of the angles along the transversal until it meets the second parallel line. It will match its corresponding angle exactly. This is known as the corresponding angle postulate: If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
We can draw a line parallel to the base of any triangle through its third vertex. Then we use transversals, vertical angles, and corresponding angles to rearrange those angle measures into a straight line, proving that they must add up to 180°.
We know that all three central angles must add together to get 360-degree, so we can subtract to find the central angle CAB. = 2 a + 2 b = 2 ( a + b ) . Therefore, the central angle measure CAB is twice the inscribed angle CDB, and this is the central angle theorem proof.
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