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### Course: Geometry (all content)>Unit 14

Lesson 8: Inscribed shapes problem solving

Sal uses the inscribed angle theorem and some algebra to prove that opposite angles of an inscribed quadrilateral are supplementary.

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• at , Sal says that in previous videos, it is shown that the measure of an inscribed angle is half the measure of the arc it intercepts. I can't find any video on Khan Academy showing this? All I can find are videos proving that the measure of an inscribed angle is half the measure of a central angle that subtends the same arc, which seems related to, but not quite the same as what he is getting at here. Could anyone point me in the direction of the video/ explain the connection between these two ideas?
• "The measure of an arc" means "the measure of the central angle that subtends that arc". It is just another way of phrasing the same thing. By default, if it is not specified, you should assume a given measure of an arc is the measure of the central angle that subtends that arc.
Also here is the lesson regarding inscribed angles and central angles, in case anyone isn't familiar with the proof behind it all:
• I came up with the proof differently. 2x°+2y°=360°. So x°+y°=180°
• Does this work even if there is no circle? That is, all quadrilaterals have this property that the opposite angles are supplementary? I'm guessing not...and that's because the four vertices have to be able to form a circle. Is this right?
• You are getting rules mixed up, some quadrilaterals have opposite angles supplementary (isosceles trapezoids, squares and rectangles because they are all 90 and many trapeziums). For parallelograms, opposite angles are congruent and consecutive angles are supplementary.
So it works when it forms a circle. If you start with a parallelogram that is not a rectangle or rhombus, you could not draw a circle that has all 4 vertices on it.
• What theorem is this called (wanted to know for proofs)?
• It's just called the Inscribed Quadrilateral Theorem.
• explain the intercepted arc please
• At , why does Sal do it as 1/2(360-2x)?
(1 vote)
• The inscribed angle theorem which was shown in the video before this one. An inscribed angle ( one that touches the circle) is related to the angle it subtends by a scale factor of two, If I know the angle, I double it to get the arc (which he did at the beginning to get 2x), and if I know the arc, I cut it in half to get the angle which is where the expression you asked about comes from. I assume you are okay with the 360-2x to be all of the circle not part of the 2x.
• When we say "the measure of the arc", are we talking about the angle?
(1 vote)
• The 'measure of an arc' or arc measure is equal to the arc length divided by the radius (s/r = 2x degrees). An arc measure is an angle the arc makes at the center of a circle, whereas the arc length is the span along the arc. In other words, an arc measure is equal to the central angle that subtends the arc.
• Is there a relationship between the length of the chord between the two endpoints of the arc and the value of the angle(with vertex at the center of the circle) ? If so, what would the graph look like(input: length of chord, output: Value of angle)? I am aware that this is not a function, but am still interested in the graph. Thanks in advance!
(1 vote)
• I believe this isn't sufficient to list a relationship (since for a larger circle, the chord will also increase in length). You will need the radius of the circle as well. After that, it's just the formula of the arc measure.
(1 vote)
• Is the measure of the arc subtended by the rays that form angle x the same as the measure of the central angle? Aren't they equal only if the radius is 1?