Topicme448ed2d8b01a89c_1528449000663_0Topic

The inscribed angleinscribed angleinscribed angle and the central anglecentral anglecentral angle

Levelme448ed2d8b01a89c_1528449084556_0Level

Third

Core curriculumme448ed2d8b01a89c_1528449076687_0Core curriculum

VIII. Plane geometry. The student:

5) applies the properties of the inscribed and the central anglecentral anglecentral angle.

Timingme448ed2d8b01a89c_1528449068082_0Timing

45 minutes

General objectiveme448ed2d8b01a89c_1528449523725_0General objective

Reasoning, including multiple‑stage arguments, giving arguments, justifying the correctness of reasoning, distinguishing a proof from an example.

Specific objectivesme448ed2d8b01a89c_1528449552113_0Specific objectives

1. Reasoning, including multiple‑stage arguments, giving arguments, justifying the correctness of reasoning, distinguishing a proof from an example.

2. Applying the inscribed angle theorem to solve problems.

3. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesme448ed2d8b01a89c_1528450430307_0Learning outcomes

The student:

- proves the inscribed angle theorem,

- applies the inscribed angleinscribed angleinscribed angle theorem.

Methodsme448ed2d8b01a89c_1528449534267_0Methods

1. Discussion.

Forms of workme448ed2d8b01a89c_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionme448ed2d8b01a89c_1528450127855_0Introduction

The teacher introduces the topic of the lesson: the proof for the inscribed angle theorem and its application to solve mathematical problems.

Students revise the definition of the central anglecentral anglecentral angle and the definition of the inscribed angleinscribed angleinscribed angle.

The teacher draws on the board a right triangletriangletriangle inscribed in a circlecirclecircle and asks the students to identify the central anglecentral anglecentral angle subtended by the same arcarcarc as the right angleangleangle of the triangletriangletriangle.

Discussion:

- What is the relationship between the measuremeasuremeasure of the inscribed angleinscribed angleinscribed angle and the measure of the central anglecentral anglecentral angle in this case?

- Can it be shown that this dependence is true for any inscribed and central anglecentral anglecentral angle subtended by the same arcarcarc?

Procedureme448ed2d8b01a89c_1528446435040_0Procedure

The teacher divides the students into 6 groups. Two groups deal with one of the following cases, trying to show that the central angleangleangle $\alpha$ measure is double the angleangleangle $\beta$ measure. Then, representatives of groups dealing with the same case check each other's reasoning, and present the agreed solution on the class forum.

[Illustration 1]

Task
The teacher shows the Slideshow to the students. It shows the next steps of the proof for case I.

[Slideshow]

The group work should be summarized with the formulation of the inscribed angleinscribed angleinscribed angle theorem.

The inscribed angle theorem.

- The measure of an inscribed angle is half the measure of the central angle subtended by the same arc. me448ed2d8b01a89c_1527752263647_0- The measure of an inscribed angle is half the measure of the central angle subtended by the same arc. 

Students work individually, solving the following problems. Having completed the exercises, they present the results and discuss them.

Task
Points A, B, C lying on a circlecirclecircle with the center S are vertices of an equilateral triangleequilateral triangleequilateral triangle. Find the measuremeasuremeasure of the central angleangleangle ASB?

Task
The sum of measures of the inscribed and the central angleangleangle subtended by the same arcarcarc equals 180°. What is the measuremeasuremeasure of the inscribed angleinscribed angleinscribed angle?

Task
The acute isosceles triangletriangletriangle with an AB base is inscribed in a circlecirclecircle with the center S. The SAB angleangleangle has 40°. What is the measuremeasuremeasure of the ACB angleangleangle?

Task
Points A, B, C, D lie on a circlecirclecircle and cut it into 4 equal arcs. What is the measuremeasuremeasure of the inscribed angleinscribed angleinscribed angle ACD?

An extra task:
The triangletriangletriangle ABC is given. The angle BAC is the right one and the angleangleangle ABC measuremeasuremeasure equals 25°. PointpointPoint E is the middle of the BC side. Find the measure of the AEC angle.

A hint: 
Enter a triangletriangletriangle in a circlecirclecircle and compare the measures of the inscribed and the central angles.

Lesson summaryme448ed2d8b01a89c_1528450119332_0Lesson summary

Students do the revision exercises.

Then together summarize the class, by formulating the inscribed angle theorem.

- The measure of an inscribed angle is half the measure of the central angle subtended by the same arc.me448ed2d8b01a89c_1527752263647_0- The measure of an inscribed angle is half the measure of the central angle subtended by the same arc.

Selected words and expressions used in the lesson plan

angleangleangle

arcarcarc

central anglecentral anglecentral angle

circlecirclecircle

equilateral triangleequilateral triangleequilateral triangle

inscribed angleinscribed angleinscribed angle

measuremeasuremeasure

pointpointpoint

triangletriangletriangle