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10.2: Inscribed Angles Theorem

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Geometry, Chapter 9, Lesson 4.

ID: 12437

Time Required: 15 minutes

Activity Overview

Students will begin this activity by looking at inscribed angles and central angles and work towards discovering a relationship among the two, the Inscribed Angle Theorem. Then, students will look at two corollaries to the theorem.

Topic: Circles

• Construct central and inscribed angles
• Inscribed angles theorem

Teacher Preparation and Notes

Associated Materials

• Student Worksheet: Inscribed Angle Theorem http://www.ck12.org/flexr/chapter/9694, scroll down to the second activity.
• Cabri Jr. Application
• INSCRIB1.8xv, INSCRIB2.8xv, INSCRIB3.8xv, INSCRIB4.8xv, and INSCRIB5.8xv

Problem 1 – Similar Triangles

Students will begin this activity by looking at inscribed angles and central angles and work towards discovering a relationship among the two.

Students will be asked to collect data by moving points \begin{align*}A\end{align*} and \begin{align*}C\end{align*}. Students are asked questions about the relationships in the circle and are asked to make a conjecture. In order to calculate the ratio of \begin{align*}m\angle{ACB}\end{align*} to \begin{align*}m\angle{ADB}\end{align*}, students can use the Calculate tool in the Actions menu.

On page 1.6, students will look at two inscribed angles intercepted by the same arc and are asked to make a conjecture about the relationship.

In an advanced setting a proof of the inscribed angle theorem and the two conjectures in problem one are appropriate and can be proved using isosceles triangles.

Problem 2 – Extension of the Inscribed Angle Theorem

In Problem 2, students will look at two more angles created from the central angle and the intercepted arc. Both sections of this problem are corollaries of the Inscribed Angle Theorem and both solutions are congruent to the measure of the central angle intercepted by the arc or one-half the measure of the central angle.

Solutions

Position Measure of \begin{align*}\angle{ACB}\end{align*} Measure of \begin{align*}\angle{ADB}\end{align*} \begin{align*}\frac{m\angle{ACB}}{m\angle{ADB}}\end{align*}
1 \begin{align*}36.65^\circ\end{align*} \begin{align*}73.30^\circ\end{align*} \begin{align*}0.5\end{align*}
2 \begin{align*}49.50^\circ\end{align*} \begin{align*}99.01^\circ\end{align*} \begin{align*}0.5\end{align*}
3 \begin{align*}49.50^\circ\end{align*} \begin{align*}99.01^\circ\end{align*} \begin{align*}0.5\end{align*}
4 \begin{align*}49.50^\circ\end{align*} \begin{align*}99.01^\circ\end{align*} \begin{align*}0.5\end{align*}

2. \begin{align*}\frac{1}{2}\end{align*}

Position Measure of \begin{align*}\angle{ACB}\end{align*} Measure of \begin{align*}\angle{AEB}\end{align*}
1 \begin{align*}36.65^\circ\end{align*} \begin{align*}36.65^\circ\end{align*}
2 \begin{align*}48.30^\circ\end{align*} \begin{align*}48.30^\circ\end{align*}
3 \begin{align*}48.30^\circ\end{align*} \begin{align*}48.30^\circ\end{align*}
4 \begin{align*}48.30^\circ\end{align*} \begin{align*}48.30^\circ\end{align*}

4. Sample answer: They are congruent.

5. diameter of the circle

6. \begin{align*}90^\circ\end{align*}

Position Measure of \begin{align*}\angle{ACB}\end{align*} Measure of \begin{align*}\angle{ADB}\end{align*} Measure of \begin{align*}\angle{AGE}\end{align*}
1 \begin{align*}36.65^\circ\end{align*} \begin{align*}73.30^\circ\end{align*} \begin{align*}36.65^\circ\end{align*}
2 \begin{align*}67.04^\circ\end{align*} \begin{align*}134.10^\circ\end{align*} \begin{align*}67.04^\circ\end{align*}
3 \begin{align*}67.04^\circ\end{align*} \begin{align*}134.10^\circ\end{align*} \begin{align*}67.04^\circ\end{align*}
4 \begin{align*}67.04^\circ\end{align*} \begin{align*}134.10^\circ\end{align*} \begin{align*}67.04^\circ\end{align*}

8. \begin{align*}\frac{1}{2}\end{align*}

Position Measure of \begin{align*}\angle{ACB}\end{align*} Measure of \begin{align*}\angle{ADB}\end{align*} Measure of \begin{align*}\angle{ABE}\end{align*}
1 \begin{align*}57.65^\circ\end{align*} \begin{align*}115.30^\circ\end{align*} \begin{align*}57.64^\circ\end{align*}
2 \begin{align*}49.20^\circ\end{align*} \begin{align*}98.40^\circ\end{align*} \begin{align*}49.19^\circ\end{align*}
3 \begin{align*}54.90^\circ\end{align*} \begin{align*}109.80^\circ\end{align*} \begin{align*}54.90^\circ\end{align*}
4 \begin{align*}63.55^\circ\end{align*} \begin{align*}127.10^\circ\end{align*} \begin{align*}63.54^\circ\end{align*}

10. \begin{align*}\frac{1}{2}\end{align*}

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Date Created:
Feb 23, 2012