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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

GeoB 6.10 Review 03

 

  • A rectangle is a parellelogram with all right angles and congruent opposite sides

GeoB 6.10 Review 04

 

A square is a rectangle with all congruent sides.

GeoB 6.10 Review 05

 


  • Circles and Triangles

GeoB 6.10 Review 01

 


  • 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°

GeoB 6.10 Review 02

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

GeoB 6.10 Incenter

  • Circumcenter

GeoB 6.10 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.
    • Quadrilaterals inscribed in circles
      • 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.

 


GeoB 6.10 Trapezoids Inscribed

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

GeoB 6.10 Q5


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