Unit 7, Lesson 8
Quadrilaterals in Circles
Integrated Math 2
8.1
Warm-up
For each quadrilateral, use a compass to see if you can draw a circle that passes through all 4 of the quadrilateral’s vertices.
A
B
C
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For each quadrilateral, use a compass to see if you can draw a circle that passes through all 4 of the quadrilateral’s vertices.
A
B
C
A
B
C
Quadrilateral is a cyclic quadrilateral.
A circle is said to be circumscribed about a polygon if all the vertices of the polygon lie on the circle. If it is possible to draw a circumscribed circle for a quadrilateral, the figure is called a cyclic quadrilateral. Not all quadrilaterals have this property.
We can prove that the opposite angles of a cyclic quadrilateral are supplementary. Consider opposite angles and , labeled and .
Angle is inscribed in the arc from to through . Angle is inscribed in the arc from to through . Together, the two arcs trace out the entire circumference of the circle, so their measures add to 360 degrees. By the Inscribed Angle Theorem, the sum of and must be half of 360 degrees, or 180 degrees. So angles and are supplementary. The same argument can be applied to the other pair of opposite angles.
When a figure is circumscribed, it is completely surrounded by another figure, so their sides, edges, vertices, or curves touch.
If a circle can be drawn around a quadrilateral so that it passes through every vertex, the quadrilateral is called a cyclic quadrilateral.
That is, a cyclic quadrilateral can be circumscribed by a circle.