Lesson 1

Lines, Angles, and Curves

  • Let’s define some line segments and angles related to circles.

1.1: Notice and Wonder: Lines and Angles

What do you notice? What do you wonder?

A photo of a round window
A picture of a bicycle wheel.  
A photo of an old wagon wheel

 

1.2: The Defining Moment

  1. The images show some line segments that are chords and some segments that are not chords.

    chords

    3 circles with lines inside each circle.

    not chords

    3 circles with lines.

    Write a definition of a chord.

  2. The images show some highlighted objects that are arcs, and some highlighted objects that are not arcs.

    arcs

    3 circles with arcs. First circle, arc along bottom of circle. Middle circle, arc along top right side of circle. Last circle, arc along top half of circle.

    not arcs

    Curve. Circle with center point and segment on lower right of circle.

    Write a definition of an arc.

  3. The images show some angles that are central angles, and some that are not.

    central angles

    3 circles with angles inside.

    not central angles

    3 circles with angles.

    Write a definition of a central angle.

1.3: Arcs, Chords, and Central Angles

The image shows a circle with 2 congruent chords.

Circle with center A. Lines DE and CB are equivalent. Arc DE orange. Arc CD blue.
  1. Draw the central angles associated with the highlighted arcs from \(D\) to \(E\) and \(B\) to \(C\).
  2. How do the measures of the 2 central angles appear to compare? Prove that this observation is true.
  3. What does this tell you about the measures of the highlighted arcs from \(D\) to \(E\) and \(B\) to \(C\)? Explain your reasoning.


Prove that the perpendicular bisector of a chord goes through the center of a circle.

Summary

Diameters and radii are 2 types of line segments that appear in circles. Here are some additional geometric objects associated with circles.

A chord is a line segment whose endpoints are on the circle. A central angle in a circle is an angle whose vertex is at the center of the circle. An arc is the portion of a circle between 2 points on the circle. The measure of an arc is defined as the measure of the central angle formed by the radii drawn to the endpoints of the arc. For example, in the image, the highlighted arc between points \(D\) and \(E\) measures 45 degrees because the central angle \(DAE\) measures 45 degrees.

chord \(GH\)

Circle with center O. Chord GH.

central angle $PQR$

Circle with center Q. Angle PQR.

arc \(DE\)

Circle center A. Diameter B C drawn. Points D and E plotted on circumference. Angle D A B, right angle. Angle D A E and C A E labeled congruent. Arc D E highlighted.

Glossary Entries

  • arc

    The part of a circle lying between two points on the circle.

  • central angle

    An angle formed by two rays whose endpoints are the center of a circle.

  • chord

    A chord of a circle is a line segment both of whose endpoints are on the circle.