Lines, Angles, and Curves
1.1: Notice and Wonder: Lines and Angles (5 minutes)
The purpose of this warm-up is to elicit ideas about different kinds of lines and angles in circles, which will be useful when students explore inscribed angles, chords, and tangent lines in later activities. While students may notice and wonder many things about these images, the idea there may be many kinds of lines and angles in circles is the important discussion point.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.
Display the images for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.
What do you notice? What do you wonder?
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
1.2: The Defining Moment (15 minutes)
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.
Arrange students in groups of 2. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.
- The images show some line segments that are chords and some segments that are not chords.
Write a definition of a chord.
The images show some highlighted objects that are arcs, and some highlighted objects that are not arcs.
Write a definition of an arc.
The images show some angles that are central angles, and some that are not.
Write a definition of a central angle.
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.” (Chords have their 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 2 points on that circle. How many arcs are defined?” (There are 2 arcs defined by any 2 points on a circle, depending on the direction we move.)
Tell students that in this unit, to distinguish between the 2 possible arcs between 2 points we will use highlighting or descriptions. It isn’t necessary to use the terms minor arc and major arc, but use them if you find them useful.
Design Principle(s): Optimize output (for comparison); Support sense-making
Supports accessibility for: Conceptual processing; Language
1.3: Arcs, Chords, and Central Angles (15 minutes)
The goal of this activity is for students to strengthen their understanding of the vocabulary terms from the previous activity. Students prove a property of congruent chords. In the process, they draw central angles and think about the relationships between radii, central angles, and arcs. The ability to understand and visualize central angles and arcs will be helpful when students study the relationship between central angles and inscribed angles in an upcoming activity.
Tell students that we define the measure of an arc as the measure of the central angle formed by drawing radii from the endpoints of the arc. Display this image for all to see:
Ask students, “What is the measure of the highlighted arc from point \(C\) to point \(D\)?” (60 degrees.) “What is the measure of the highlighted arc from point \(D\) to point \(B\)?” (120 degrees.) “What is the measure of the arc that goes from point \(C\) to point \(B\) through point \(D\)?” (180 degrees.) “What is the measure of the other arc that goes from point \(C\) to point \(B\) ?” (It also measures 180 degrees.)
Supports accessibility for: Visual-spatial processing
The image shows a circle with 2 congruent chords.
- Draw the central angles associated with the highlighted arcs from \(D\) to \(E\) and \(B\) to \(C\).
- How do the measures of the 2 central angles appear to compare? Prove that this observation is true.
- What does this tell you about the measures of the highlighted arcs from \(D\) to \(E\) and \(B\) to \(C\)? Explain your reasoning.
Are you ready for more?
Prove that the perpendicular bisector of a chord goes through the center of a circle.
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 2 triangles? Alternatively, ask them 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 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.
Sample response: We can prove that chords \(DE\) and \(BC\) are congruent, and the arcs that go with them have the same measures. The proofs are the same in that the segments \(AB,AC,AD,\) and \(AE\) are congruent because they are radii. But now it’s a given that angles \(DAE\) and \(BAC\) are congruent, so the triangles are congruent by the Side-Angle-Side Triangle Congruence Theorem. Now the 2 chords are congruent because they’re corresponding parts of congruent triangles.
The goal is to practice interpreting diagrams that include chords, arcs, and measures of central angles. Display this image for all to see:
Tell students that segment \(CD\) is a diameter. Then ask:
- “How many chords are drawn in this picture? Name them all.” (There are 3 chords. Segments \(BC,DE,\) and \(CD\) are all chords.)
- “Is angle \(CBA\) a central angle? Why or why not?” (No. The vertex of angle \(CBA\) is point \(B\), which is not the center of the circle.)
- “What arc and angle measures can you find?” (The chords \(BC\) and \(DE\) are congruent. This means that angles \(BAC\) and \(EAD\) are congruent because these central angles are created by drawing radii from the endpoints of the chords. So angle \(EAD\) measures 50 degrees, as does the arc between \(B\) and \(C\) as well as the arc between \(D\) and \(E\). The measure of the highlighted arc from \(B\) to \(E\) is 80 degrees because central angle \(BAE\) measures 80 degrees.)
1.4: Cool-down - Draw It (5 minutes)
Student Lesson 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.