AAS - Angles, Arcs, and Segments Module Overview

Math_GeoConceptsConnectBanner.png Angles, Arcs, and Segments Introduction

image of a supermoon with caption: "supermoon: November 14, 2016 The next supermoon will not occur until November 25, 2034 and we haven't seen the supermoon since January 26, 1948. That's one big circle there!"Take one minute and name as many circular objects as you can think of. How many did you come up with? Most people can name many circular objects off the top of their head, but few can list the many properties and special parts of a circle. With circles appearing so often in everyday life, it is important and helpful for us to understand their special properties. It is these same properties that allow us to solve many every day problems. What is the best location for a cell tower that is supposed to service three different cities? What are the different types of intersections the sun and the horizon may have? What is the circumference of a broken plate that we only have a few pieces of? These are just a few questions that can be answered using circles and their properties.

Essential Questions

  • How do we know all circles are similar?
  • How can we apply circle properties to real life situations in order to solve problems?
  • What are angles, radii, chords, tangents, and secants and how are they related?
  • What is the difference between central, inscribed, and circumscribed angles?
  • How can we show that an inscribed angle on a diameter is a right angle?
  • How do we know that a radius of a circle is perpendicular to a tangent at the point where they both intersect the circle?
  • What is the difference between an inscribed and circumscribed circle of a triangle, and how can we construct each?
  • How can we prove the various properties of angles of a quadrilateral inscribed in a circle?
  • What is the method for constructing a tangent line from a point outside the circle to a point on the circle?

Key Terms

Arc - a portion of the circumference of a circle

Arc measure - the measure of the angle at the center of the circle that creates an arc

Major arc - an arc whose measure is greater than 180 degrees

Minor arc - an arc whose measure is less than 180 degrees

Point of tangency - the point where a tangent line intersects the circle

Chord - a segment whose endpoints both lie on the circle

Central angle - an angle whose vertex is the center of the circle

Common tangent - a line that is tangent to two coplanar circles

Cyclic quadrilaterals - a quadrilateral inscribed in a circle (all 4 vertices lie on the circle)

Inscribed angle - an angle whose vertex is on the circle and whose sides are made up by chords of a circle

Circumscribed angle - an angle whose vertex is outside the circle and whose sides are made by two intersecting tangent lines to a circle 

Circumscribed circle - a circle that passes through all the vertices of a polygon

Inscribed polygon - a polygon contained inside a circle whose vertices all lie on the circle

Inscribed circle - a circle contained in a polygon where each side of the polygon is tangent to the circle

Secant line - a line that intersects a circle at exactly two points

Semicircle - a half circle; created by an arc whose measure is equal to 180 degrees

Tangent line - a line in the plane of the circle that intersects it at exactly one point

Circumcenter - the point of intersection of the perpendicular bisectors of a triangle that is inscribed in a circle; this is also the center of the circle circumscribed around a given triangle

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