# 6.10 – Circles and Polygons

## Key Terms

No key terms for this section.

## Review

 A polygon is a closed plane figure with all straight sides.   A parallelogram is a quadrilateral with parallel opposite sides   A rectangle is a parellelogram with all right angles and congruent opposite sides   A square is a rectangle with all congruent sides.   Circles and Triangles   Angles Inscribed angle – An angle formed by two chords that share an endpoint on a circle. It is half the measure of the intercepted arc. Supplementary angles – Two angles whose measures have the sum of 180°. Complementary angles – Two angles whose measures have the sum of 90°

## Notes

Incenters and Circumcenters of Polygons
• The ideas of incenter and circumcenter for all polygons are the same as what you have seen for triangles.
• Incenter

• Circumcenter

Inscribed and Circumscribed Circles
• Circle Inscribed in a Polygon
• The circle is inside the polygon and tangent to each side of the polygon.
• Circle Circumscribed About a Polygon
• The circle is wrapped around the outside of the polygon and intersects each of the polygon’s vertices.
• The measures of opposite angles of a quadrilateral (4 sided polygon) add up to 180° (supplementary).
• The ONLY parallelogram that can be inscribed in a circle is a rectangle.
• All squares are rectangles.

## Examples

 Ex 1. In the diagram below, $\odot{P}$ is circumscribed about quadrilateral ABCD. What is the value of x? Since opposite angles are supplementary, x + 120 = 180. Subtract 120 on both sides: x = 60