In this activity, students write definitions of chords, arcs, and central angles. As students work, if they first propose less formal or imprecise language, invite them to reword their definitions with more precise language (MP6).  

The goal is to summarize the definitions of these terms. Ask several students to share their definitions, and invite the class to discuss similarities and differences in these definitions. Here are additional questions for discussion:

• “Compare and contrast chords, diameters, and radii.” (A chord has its endpoints on the circle. A diameter is a special kind of chord—it passes through the center of the circle. A radius is the only one that does not have both endpoints on the circle. Instead, one of a radius’s endpoints is at the center of the circle.)
• “Would we consider a complete circle to be an arc?” (Yes, a complete circle does fit our definition of an arc.)
• “What kind of segments make up a central angle?” (Radii. Technically, the angle is formed by rays that contain the radii.)
• “Draw a circle and draw two points on that circle. How many arcs are defined?” (There are two arcs defined by any two points on a circle: one arc in each direction of the circle.)

Tell students that, in this unit, to distinguish between the two possible arcs between two points, we will use highlighting or descriptions. It isn’t necessary to use the terms “minor arc” and “major arc,” but it may be useful.

If students struggle to write a proof, ask them if any triangle congruence theorems might apply. What do we know about any of the side lengths or angles in the two triangles? Alternatively, ask students if there is a sequence of rigid motions that will take one triangle onto the other.

The goal is to prove the converse of what was proved in the activity. Display this image for all to see:

• Ask students, “Suppose we start out not knowing anything about the chord lengths but knowing that the central angles are congruent. What can we prove, and how does the proof differ from the one in the activity?” Give students 1–2 minutes of quiet work time and then time to share their thoughts with a partner. Follow with a whole-class discussion. Points for discussion: We can prove that chords and are congruent and that the arcs that go with them have the same measures. 
• The proofs are the same in that the segments and are congruent because they are radii. But now it’s a given that angles and are congruent, so the triangles are congruent by the Side-Angle-Side Triangle Congruence Theorem. Now the two chords are congruent because they’re corresponding parts of congruent triangles.