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# 10.2: Inscribed Angle Theorem

Difficulty Level: At Grade Created by: CK-12

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

## Problem 1 – Inscribed Angle Theorem

Start the Cabri Jr. application by pressing the APPS key and selecting CabriJr. Open the file INSCRIB1 by pressing Y=\begin{align*}Y=\end{align*}, selecting Open..., and selecting the file. In INSCRIB1, you are given circle D\begin{align*}D\end{align*} with radius AD\begin{align*}AD\end{align*}. Angle ADB\begin{align*}ADB\end{align*} is a central angle and ACB\begin{align*}\angle{ACB}\end{align*} is an inscribed angle.

1. Move point A\begin{align*}A\end{align*} to 2 different positions and point C\begin{align*}C\end{align*} to 2 different positions and collect the data in the table below. Calculate the ratios of mACB\begin{align*}m\angle{ACB}\end{align*} to mADB\begin{align*}m\angle{ADB}\end{align*} for each position and record the calculation in the table below.

Position Measure of ACB\begin{align*}\angle{ACB}\end{align*} Measure of ADB\begin{align*}\angle{ADB}\end{align*} mACBmADB\begin{align*}\frac{m\angle{ACB}}{m\angle{ADB}}\end{align*}
1
2
3
4

2. Angles ACB\begin{align*}ACB\end{align*} and ADB\begin{align*}ADB\end{align*} are said to intercept the same arc, AB\begin{align*}AB\end{align*}, because they go through the same points A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} on the circle. An inscribed angle in a circle is __________ the measure of the central angle that intercepts the same arc on the circle.

Open the file INSCRIB2. You are given circle D\begin{align*}D\end{align*}. Angles ACB\begin{align*}ACB\end{align*} and AEB\begin{align*}AEB\end{align*} are inscribed angles and intercept the same arc.

3. Move point A\begin{align*}A\end{align*} to 2 different positions and move point E\begin{align*}E\end{align*} to 2 different positions and collect the data in the table below.

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

4. Make a conjecture about two inscribed angles who intercept the same arc in a circle.

Open the file INSCRIB3. You are given circle D\begin{align*}D\end{align*}. Use this file to answer the following questions.

5. In circle D\begin{align*}D\end{align*}, what kind of segment is AB\begin{align*}AB\end{align*}?

6. In circle D\begin{align*}D\end{align*}, what is mACB\begin{align*}m\angle{ACB}\end{align*}? (Hint: Use your answer to Exercise 4 to help you.).

## Problem 2 – Extension of the Inscribed Angle Theorem

Open the file INSCRIB4. You are given circle D\begin{align*}D\end{align*}, AB\begin{align*}AB\end{align*}, and ACB\begin{align*}\angle{ACB}\end{align*}. Point G\begin{align*}G\end{align*} is a point on AB\begin{align*}AB\end{align*}, ACB\begin{align*}\angle{ACB}\end{align*} is an inscribed angle, and AG\begin{align*}AG\end{align*} and BG\begin{align*}BG\end{align*} are lines.

7. Move point A\begin{align*}A\end{align*} to 2 different positions and move point G\begin{align*}G\end{align*} to 2 different positions and collect the data in the table below.

Position Measure of ACB\begin{align*}\angle{ACB}\end{align*} Measure of ADB\begin{align*}\angle{ADB}\end{align*} Measure of AGE\begin{align*}\angle{AGE}\end{align*}
1
2
3
4

8. Make a conjecture: The angle formed by the intersection of AG\begin{align*}\overrightarrow{AG}\end{align*} and BG\begin{align*}\overrightarrow{BG}\end{align*} is _______ the measure of the central angle ADB\begin{align*}ADB\end{align*}.

Open the file INSCRIB5. You are given circle D\begin{align*}D\end{align*}, arc AB\begin{align*}AB\end{align*}, and ACB\begin{align*}\angle{ACB}\end{align*}. Point G\begin{align*}G\end{align*} is a point on arc AB\begin{align*}AB\end{align*} and \begin{align*}\angle{ACB}\end{align*} is an inscribed angle. Also, you are given chord \begin{align*}AB\end{align*} and a tangent line \begin{align*}BE\end{align*}.

9. Move point \begin{align*}A\end{align*} to 2 different positions and move point \begin{align*}B\end{align*} to 2 different positions and collect the data in the table below.

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
2
3
4

10. Make a conjecture: The angle between a chord and the tangent line at one of its intersection points equals ______________ of the central angle intercepted by the chord.

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