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When a quadrilateral is inscribed in a circle, the opposite angles are supplementary to one another.

Angles A and C  and angles D and B are supplementary. (add to 180)
opposite angles of an inscribed quadrilateral are supplementary

Polygon is NOT inscribed. All vertices do NOT touch the circle.

inscribed angle and a circle
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Inscribed Angles

Inscribed Angle Formula

What is an inscribed angle?

 inscribed Circle
Figure A
Figure B

Inscribed Polygon

In this example, the Polygon is inscribed in the circle (all vertices touch the circle.)
The Circle is circumscribed about the polygon.
The Polygon is circumscribed about the circle and the Circle is inscribed in the polygon. 

 inscribed Circle
Polygon not inscribed
Figure C
A circle can also have inscribed angles, which are formed when two secants intersect on the circle. The point in which they intersect is the vertex.

An inscribed angle can also be thought of as two chords sharing an end point. 

Remember a chord is a line segment in which the endpoints are on the circle and a secant is a line that intersects a circle at two points.

The angle measure of the inscribed angle can be calculated using the following formula:
Ex.  If measure of arc AB is 80 degrees, then m<C = 40 degrees.

Inscribed Polygon

Circumscribed Polygon

horizontal line
inscribed angle of circle
chord of a circle
Inscribed polygon is a polygon inside a circle in which all of the vertices touch the circumference of the circle. 

Vertices (plural of vertex) is the point where two or more straight lines meet and create a corner.

Let's look at some examples of Inscribed and Circumscribed figures. 
An object drawn around another geometric figure that touches the geometric figure at every vertices or each side is said to be a circumscribed figure.
The vertices are highlighted
Inscribed angle = 1/2 x measure of the Inscribed arc.
Inscribed Angle
Inscribed arc
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Hi welcome to MooMooMath. Today we are looking at inscribed angles when talking about polygons and circles. First we have a triangle and it is going to be inscribed in the circle. So I will draw the circle around the triangle so we say the triangle is inscribed in the circle. Now what if the circle is inscribed in the polygon? That just means it will be on the inside. So let me draw a polygon. I have a quadrilateral around the circle so we say we have a circle inscribed in the polygon. In order to be inscribed all the vertices need to touch the circle and the circle has to be tangent to the polygon. These are points of tangents so they touch in one point. Now let’s look at circumscribed. If this polygon is circumscribed about the circle that means the circle is inside of it so we say the polygon is circumscribed about the circle. If the circle is circumscribed about the polygon inside the circle, it can be any sized polygon in this case I have a hexagon inside it) so I would say the circle is circumscribed about the polygon. So that is the difference between inscribed and circumscribed. So a polygon inscribed in a circle means the polygon is inside. A circle that is inscribed in the polygon is on the inside. Hope this video was helpful.