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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 Cabri Jr. Open the file INSCRIB1 by pressing $Y=$, selecting Open..., and selecting the file. In INSCRIB1, you are given circle $D$ with radius $AD$. Angle $ADB$ is a central angle and $\angle{ACB}$ is an inscribed angle.

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

Position Measure of $\angle{ACB}$ Measure of $\angle{ADB}$ $\frac{m\angle{ACB}}{m\angle{ADB}}$
1
2
3
4

2. Angles $ACB$ and $ADB$ are said to intercept the same arc, $AB$, because they go through the same points $A$ and $B$ 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 $D$. Angles $ACB$ and $AEB$ are inscribed angles and intercept the same arc.

3. Move point $A$ to 2 different positions and move point $E$ to 2 different positions and collect the data in the table below.

Position Measure of $\angle{ACB}$ Measure of $\angle{AEB}$
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 $D$. Use this file to answer the following questions.

5. In circle $D$, what kind of segment is $AB$?

6. In circle $D$, what is $m\angle{ACB}$? (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 $D$, $AB$, and $\angle{ACB}$. Point $G$ is a point on $AB$, $\angle{ACB}$ is an inscribed angle, and $AG$ and $BG$ are lines.

7. Move point $A$ to 2 different positions and move point $G$ to 2 different positions and collect the data in the table below.

Position Measure of $\angle{ACB}$ Measure of $\angle{ADB}$ Measure of $\angle{AGE}$
1
2
3
4

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

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

9. Move point $A$ to 2 different positions and move point $B$ to 2 different positions and collect the data in the table below.

Position Measure of $\angle{ACB}$ Measure of $\angle{ADB}$ Measure of $\angle{ABE}$
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.

## Date Created:

Feb 23, 2012

Nov 03, 2014
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