Topicma4a714ff03787e92_1528449000663_0Topic

Angles in circles

Levelma4a714ff03787e92_1528449084556_0Level

Third

Core curriculumma4a714ff03787e92_1528449076687_0Core curriculum

VIII. Plane geometry. The student:

5) applies the properties of inscribed and central angles.

Timingma4a714ff03787e92_1528449068082_0Timing

45 minutes

General objectivema4a714ff03787e92_1528449523725_0General objective

Using and interpreting the representation. Using mathematical objects and manipulating them, interpreting mathematical concepts.

Noticing regularities, similarities and analogies and formulating relevant conclusions.

Specific objectivesma4a714ff03787e92_1528449552113_0Specific objectives

1. Identifying central and inscribed angleangleangle in a circle.

2. Applying basic properties of inscribed and central angles.

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

Learning outcomesma4a714ff03787e92_1528450430307_0Learning outcomes

The student:

- identifies central and inscribed angleinscribed angleinscribed angle of a circle,

- applies the basic properties of inscribed and central angles.

Methodsma4a714ff03787e92_1528449534267_0Methods

1. Observation.

2. Brainstorming.

Forms of workma4a714ff03787e92_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionma4a714ff03787e92_1528450127855_0Introduction

The teacher introduces the topic of the lesson: analysing angles of a circle and their properties.

Task
The teacher asks the students to draw a circle and a few angles with vertices lying on the circle or inside the circle.

Then, students discuss in pairs how these angles could be classified, e.g. by position of the angle’s vertex. The brainstorming phase should be summarized with the introduction of the definitions of the central and the inscribed angleinscribed angleinscribed angle.

Procedurema4a714ff03787e92_1528446435040_0Procedure

Definition – the central anglecentral anglecentral angle.

- The central angle is an angle whose vertex is at the center of the circle.ma4a714ff03787e92_1527752263647_0- The central angle is an angle whose vertex is at the center of the circle.

[Illustration 1]

The angleangleangle ASC is subtended by the arc AC. The arcarcarc AC is intercepted by the angle ASC.

Points A and C cut the circle into two arcs. The AC arcarcarc is a part of the circle running from point A to point C in a counter‑clockwise direction. Arcs AC and CA are different.

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

Task
- The central anglecentral anglecentral angle ASC of a measuremeasuremeasure $\alpha$ is subtended by the arcarcarc AC. Find the measure of the central angle subtended on the arc CA.
- What is the largest possible measure of a central angle?
- The central anglecentral anglecentral angle intercepts an arcarcarc whose length is equal to $\frac{1}{9}$ of the circle's circumferencecircumferencecircumference. What is the measuremeasuremeasure of this angleangleangle?

An extra task:
Let n be a positive natural number. What is the measuremeasuremeasure of the center anglecentral anglecenter angle that intercepts an arcarcarc whose length is equal to $\frac{1}{n}$ of the circumferencecircumferencecircumference of the circle?

Definition – the inscribed angleinscribed angleinscribed angle.

- The inscribed angle is an angle whose vertex lies on the circle and whose arms are two secant lines.inscribed angle- The inscribed angle is an angle whose vertex lies on the circle and whose arms are two secant lines.

[Illustration 2]

The angle ABC is subtended by the arc AC. The arc AC is intercepted by the angleangleangle ABC. The vertex B does not lie on this arcarcarc.

Students work individually or in pairs, using computers. They observe how the measure of the inscribed angleinscribed angleinscribed angle changes depending on the length of the intercepted arcarcarc. Having completed the exercise, they draw conclusions.

Task
Change the position of the point on a circle and observe how the measuremeasuremeasure of the inscribed angleinscribed angleinscribed angle changes. Write conclusions based on your observations.

[Geogebra applet]

Conclusion:

- The longer the arc, the greater the measure of the inscribed angle.
- The inscribed angles subtended by the same arc have equal measures.ma4a714ff03787e92_1527712094602_0- The longer the arc, the greater the measure of the inscribed angle.
- The inscribed angles subtended by the same arc have equal measures.

Students work individually and then they discuss the results.

Task
A square ABCD is inscribed in a circle. Find the length of the arc that is intercepted by the inscribed angleinscribed angleinscribed angle ABC? Generalize your observations. Remember that inscribed angles subtended by the same arcarcarc have equal measures.

Conclusion:

- An inscribed angleinscribed angleinscribed angle subtended by a semicircle is a right angleright angleright angle.

This property of angles subtended by a semicircle can be formulated differently: a triangle inscribed in a circle, for which one of the sides is the diameterdiameterdiameter of the circle, is the right triangle. The diameterdiameterdiameter of the circle is the hypotenuse of this triangle.

Task
Highlight the arcs subtended by the angles ASC and ABC.

[Illustration 3]

An extra task:
In a regular hexagon ABCDEF, the ACF angleangleangle between the diagonals AC and CF is 30°. What is the measuremeasuremeasure of the angle CFD between the diagonals CF and DF?

A hint:
Enter a hexagon in a circle and consider the angles ACF and CFD as angles inscribed in a circle.

Lesson summaryma4a714ff03787e92_1528450119332_0Lesson summary

Students do the revision exercises.

Then together summarize the class, by formulating the conclusions to memorize.

- The central anglecentral anglecentral angle is an angleangleangle whose vertex is at the center of the circle.

- An inscribed angleinscribed angleinscribed angle is an angle whose vertex lies on the circle and whose arms are two secant lines.

- Inscribed angles subtended by the same arcarcarc on the circle have equal measures.

- An inscribed angle subtended by a semicircle is the right angleright angleright angle.

Selected words and expressions used in the lesson plan

angleangleangle

arcarcarc

central anglecentral anglecentral angle

circumferencecircumferencecircumference

diameterdiameterdiameter

inscribed angleinscribed angleinscribed angle

measuremeasuremeasure

right angleright angleright angle

secant linesecant linesecant line