Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
The purpose of this Warm-up is to elicit the idea that two chords form an angle that’s not a central angle, which will be useful when students investigate properties of inscribed angles in a subsequent activity. While students may notice and wonder many things about these images, the fact that the inscribed angle is formed by two chords and its vertex is on the circle is the most 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.
Launch
Display the image 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 with their partner the things they notice and wonder.
Activity
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Student Task Statement
What do you notice? What do you wonder?
Student Response
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Activity Synthesis
The goal of this discussion is for students to learn the definition of “inscribed angle.” Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
Tell students that an inscribed angle in a circle is an angle formed by two chords that share an endpoint. An inscribed angle’s vertex lies on the circle. The other two endpoints of the chords define an arc. When we talk about the arc that an inscribed angle defines, we are referring to the arc that does not pass through the vertex of the inscribed angle.
In this lesson, students conjectured that the measure of an inscribed angle is half the measure of the central angle that defines the same arc. Tell students that in the unit “Coordinate Geometry,” they investigated a special case of this theorem that they will revisit now.
Invite students to use a compass to draw a circle. Then they should place an index card so a corner lies on the circle and the sides of the card form an inscribed angle. Students should trace the index card to draw this angle, then connect the points where the chords intersect the circle. Ask students what they notice about this third line, and why does this happen? (It is a diameter. It goes through the center of the circle. The index card created a 90-degree inscribed angle, so the central angle must measure twice that, or 180 degrees. A central angle that is a straight line is a diameter.)
Remind students that in the unit “Coordinate Geometry,” we started the other way around, by drawing a diameter. Ask students, “What does the Inscribed Angle Theorem tell us about any inscribed angle that defines a 180-degree arc?” (It must measure 90 degrees.)
Student Lesson Summary
We have discussed central angles such as angle . Another kind of angle in a circle is an inscribed angle, or an angle formed by two chords that share an endpoint. In the image, angle is an inscribed angle.
It looks as though the inscribed angle is smaller than the central angle that defines the same arc. In fact, the measure of an inscribed angle is always exactly half the measure of the associated central angle. For example, if the central angle measures 50 degrees, the inscribed angle must measure 25 degrees, even if we move point along the circumference (without going past or ). This also means that all inscribed angles that define the same arc are congruent.
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In this activity, students identify and describe the relationship between central and inscribed angles that define the same arc.
In the digital version of the activity, students use an applet to visualize the relationship between these angles before making a conjecture. The applet allows students to move the points that create a central and inscribed angle that define the same arc, and to observe how the angles’ measures change.
This activity works best when each student has access to devices that can run the applet because students will benefit from seeing the relationship in a dynamic way. If students don't have individual access, projecting the applet would be helpful during the Activity Synthesis.
Launch
Arrange students in groups of 4.
MLR2 Collect and Display. Circulate to listen for and collect the language that students use as they describe what they notice about the measure of angle . On a visible display, record words and phrases, such as “half,” “smaller than the central angle,” and “our group’s measures are the same.” Invite students to borrow language from the display as needed, and update it throughout the lesson. For example, a student may rephrase “The angle is half” as “The measure of angle is half the measure of angle .” This will help students use the mathematical language necessary to describe the relationship between an inscribed angle and a central angle that defines the same arc. Advances: Conversing, Reading
Activity Synthesis
The goal of this discussion is for students to understand the Inscribed Angle Theorem.
Ask students to share their conjectures. Press them to attend to precision in the language they use. Next, display these images for all to see:
Ask students to compare and contrast these images. (All three images have a central angle and an inscribed angle . In the first case the central angle is less than 180 degrees, in the second it’s exactly 180 degrees, and in the third it’s more than 180 degrees. In all three cases, the inscribed angle is half the measure of the central angle, so this supports our conjecture.) Do the ideas support or contradict the class conjectures?
Add the following assertion to the class reference chart, and ask students to add it to their reference charts. Tell students that this can be proven to be true, but since we haven’t done so in class, we will add it as an assertion:
Inscribed Angle Theorem: The measure of an inscribed angle is half the measure of the central angle that defines the same arc. (Assertion)
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Activity Narrative
In this activity, students apply their understanding of relationships between arcs, central angles, and inscribed angles to prove that if two chords and intersect at point , then triangles and are similar. Students prove a specific case in the activity and then generalize in the Activity Synthesis.
In this partner activity, students take turns sharing their initial ideas and first drafts. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). As students revise their writing, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6).
This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.
Launch
Give students 3–4 minutes of work time. If necessary after that time, pull the class together to ensure students are able to make connections between the given measures of the arcs and the measures of the angles in the triangles.
Activity
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Student Task Statement
The image shows a circle with chords and . The highlighted arc from point to point measures 100 degrees. The highlighted arc from point to point measures 140 degrees.
Prove that triangles and are similar.
Student Response
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Building on Student Thinking
Students may assume that is the center of the circle. Ask these students to mark a point that appears to be the center of the circle and label it .
If students struggle to connect the arc measurements to the inscribed angle measurements, suggest they mark the center of the circle and then draw a central angle using radii that intersect the circle at points and . Then suggest they look at their work from the previous activity and try to find connections.
Activity Synthesis
The goal of this discussion is to generalize to all pairs of intersecting chords.
Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to to prove that triangles and are similar. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner’s ideas and writing.
If time allows, display these prompts for feedback:
“Your explanation tells me .”
“Can you say more about ?”
“A detail (or word) you could add is , because .”
Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.
“How many pairs of angles do you need to show are congruent in order to show the triangles are similar?” (We need to show that two pairs are congruent. The third pair of angles must then be congruent because the angle measures in a triangle always add to 180 degrees.)
“What if we didn’t have the measures of the arcs, but we just had the diagram? Could we still prove the triangles are similar?” (Yes. Angles and are congruent no matter what the measures of the arcs are because they are vertical angles. And angles and are inscribed in the same arc, so they will always be the same measure—half of that arc.)
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, color code to indicate that angles and are both inscribed in the same arc, as are angles and . Display two copies of the image so that this can be clearly seen. Supports accessibility for: Visual-Spatial Processing
Standards Alignment
Building On
Addressing
HSG-C.A.2
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Use a protractor to find the approximate degree measure of angle .
Mark a point on the circle that is not on the highlighted arc from to . Each member of your group should choose a different location for point . Draw chords and . Use a protractor to find the approximate degree measure of angle .
Share your results with your group. What do you notice about your answers?
Make a conjecture about the relationship between an inscribed angle and the central angle that defines the same arc.
