# Lesson 2

Inscribed Angles

## 2.1: Notice and Wonder: A New Angle (5 minutes)

### Warm-up

The purpose of this warm-up is to elicit the idea that 2 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 2 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 the things they notice with their partner, followed by a whole-class discussion.

### Student Facing

What do you notice? What do you wonder?

### Student Response

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### Activity Synthesis

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.

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 will refer to the particular arc that does not pass through the vertex of the inscribed angle.

## 2.2: A Central Relationship (15 minutes)

### Activity

In this activity, students identify and describe the relationship between central and inscribed angles that define the same arc.

This activity works best when each student has access to devices that can run the GeoGebra 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 synthesis.

### Launch

Tell students to explore the applet and answer the questions without the angle measures shown until instructed in the activity to show these measures.

*Conversing: MLR2 Collect and Display.*As students work on this activity, listen for the language students use to describe what they notice about the measure of angle \(QBC\). Write the students’ words and phrases on a visual display. Invite students to use the display to revise and improve on how ideas are communicated. For example, a statement such as, “The angle is half” can be improved with the statement, “The measure of angle \(QBC\) is half the measure of angle \(QAC\).” 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.

*Design Principle(s): Optimize output (for generalization); Maximize meta-awareness*

### Student Facing

Use the applet to answer the questions. Do not show the angle measures until you are told to.

- Name the central angle in this figure.
- Name the
**inscribed angle**in this figure. - Move point \(B\) around the circle. As you move this point, what happens to the measure of angle \(QBC\)? Show the angle measures to confirm.
- Move points \(C,Q,\) and \(B\) to new positions. Record the measure of angles \(QAC\) and \(QBC\). Repeat this several times.
- Make a conjecture about the relationship between an inscribed angle and the central angle that defines the same arc.

### Student Response

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### Launch

Arrange students in groups of 4.

*Conversing: MLR2 Collect and Display.*As students work on this activity, listen for the language students use to describe what they notice about the measure of angle \(QBC\). Write the students’ words and phrases on a visual display. Invite students to use the display to revise and improve on how ideas are communicated. For example, a statement such as, “The angle is half” can be improved with the statement, “The measure of angle \(QBC\) is half the measure of angle \(QAC\).” 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.

*Design Principle(s): Optimize output (for generalization); Maximize meta-awareness*

### Student Facing

Here is a circle with central angle \(QAC\).

- Use a protractor to find the approximate degree measure of angle \(QAC\).
- Mark a point \(B\) on the circle that is
*not*on the highlighted arc from \(C\) to \(Q\). Each member of your group should choose a different location for point \(B\). Draw chords \(BC\) and \(BQ\). Use a protractor to find the approximate degree measure of angle \(QBC\). - 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.

### Student Response

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### Student Facing

#### Are you ready for more?

Here is a special case of an inscribed angle where one of the chords that defines the inscribed angle goes through the center. The central angle \(DCF\) measures \(\theta\) degrees, and the inscribed angle \(DEF\) measures \(\alpha\) degrees. Prove that \(\alpha=\frac12 \theta\).

### Student Response

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### Activity Synthesis

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. Do they support or contradict the class conjectures? (Sample responses: All 3 images have a central angle \(CAQ\) and an inscribed angle \(CBQ\). 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 3 cases, the inscribed angle is half the measure of the central angle, so this supports our conjecture.)

Ask students to add this assertion to their reference charts as you add it to the class reference chart. 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*)

## 2.3: Similarity Returns (15 minutes)

### Activity

In this activity, students apply their understanding of relationships between arcs, central angles, and inscribed angles to prove that if 2 chords \(BC\) and \(DE\) intersect at point \(F\), then triangles \(CFD\) and \(EFB\) are similar. Students prove a specific case in the activity, then generalize in the synthesis.

### 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.

*Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.*Use this routine to help students improve their written response to prove that triangles \(CFD\) and \(EFB\) are similar. At the appropriate time, give students time to meet with 2–3 partners to share and get feedback. Display prompts that students can use to help their partner strengthen and clarify their ideas. For example, "Your explanation tells me . . .", "Can you say more about . . . ?", and "A detail (or word) you could add is _____, because . . . ." Give students with 3–4 minutes to revise their initial draft based on the feedback they received. This will help students use precise language to prove that the triangles formed by intersecting chords are similar.

*Design Principle(s): Optimize output (for justification); Cultivate conversation*

### Student Facing

The image shows a circle with chords \(CD,CB,ED,\) and \(EB\). The highlighted arc from point \(C\) to point \(E\) measures 100 degrees. The highlighted arc from point \(D\) to point \(B\) measures 140 degrees.

Prove that triangles \(CFD\) and \(EFB\) are similar.

### Student Response

### Anticipated Misconceptions

Students may assume that \(F\) is the center of the circle. Ask these students to mark in a point that appears to be the center of the circle and label it \(A\).

If students struggle to connect the arc measurements to the inscribed angle measurements, suggest they mark the center of the circle, then draw a central angle using radii that intersect the circle at points \(C\) and \(E\). Then suggest they look at their work from the previous activity and try to find connections.

### Activity Synthesis

The goal is to generalize to all pairs of intersecting chords. Ask students:

- “How many pairs of angles do you need to show are congruent in order to show the triangles are similar?” (You only need to show that 2 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 \(CFD\) and \(EFB\) are congruent no matter what the measures of the arcs are because they are vertical angles. And angles \(CBE\) and \(CDE\) 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, use color to indicate that angles \(CDE\) and \(CBE\) are both inscribed in the same arc, as are angles \(DCB\) and \(DEB\). Display two copies of the image so that this can be clearly seen.

*Supports accessibility for: Visual-spatial processing*

## Lesson Synthesis

### Lesson Synthesis

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 Coordinate Geometry unit 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. What do 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 Coordinate Geometry unit, 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.)

## 2.4: Cool-down - Inscribed Angle Measures (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

We have discussed central angles such as angle \(AOB\). Another kind of angle in a circle is an **inscribed angle**, or an angle formed by 2 chords that share an endpoint. In the image, angle \(ACB\) 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 \(AOB\) measures 50 degrees, the inscribed angle \(ACB\) must measure 25 degrees, even if we move point \(C\) along the circumference (without going past \(A\) or \(B\)). This also means that all inscribed angles that define the same arc are congruent.