Central angle- an angle with a vertex at the center of the circle. The central angle is equal to the degree measure of the arc on which it rests . Inscribed angle- an angle whose vertex lies on a circle and both sides intersect this circle

Central angle

This is an angle with its vertex at the center of the circle.

Arc of a circle corresponding to the central angle

This is the part of the circle located inside the corner

Degree measure of a circular arc

This is the degree measure of the corresponding central angle.

=  AOB

Inscribed angle

This is an angle whose vertex lies on a circle and whose sides intersect the circle.

Proofs of theorems about angles associated with a circle Theorem 1 . Magnitude inscribed angle equal to half the value central angle, resting on the same arc. Proof . Let us first consider the inscribed angle ABC, side B.C. which is diameter circle, and central angle AOC

Since the segments A.O. And B.O. are radii of the circle, That triangle AOB – isosceles, and the angle ABO equal to angle OAB . Because the angle AOC is external angle of a triangle AOB, then the equalities are true

Thus, in the case when one of the sides of the inscribed angle passes through the center of the circle, Theorem 1 is proven.

Now consider the case when the center of the circle lies inside the inscribed angle.

and Theorem 1 is proven in this case.

It remains to consider the case when the center of the circle lies outside the inscribed angle

In this case the equalities are true

which completes the proof of Theorem 1.

Description:

This presentation is a multimedia teaching aid intended for school geometry lessons.

All information collected here is clearly illustrated with accessible examples in the form of drawings, which contribute to optimal development and understanding of the topic.

The purpose of this lesson is to introduce the concepts of inscribed and central angles. Students also become familiar with the properties inherent in an inscribed angle and the consequences that follow from them.

The material presented here is presented in understandable language, it is optimally adapted for quick comprehension by school-level students, while maintaining the accuracy and rigor of logical formulations.

The work will provide an opportunity for students to become familiar with relevant concepts, as well as repeat the basic types of angles. In addition, they will be able to understand the proofs of the properties of an angle inscribed in a circle, after which they will be able to obtain the necessary consequences from this theorem. They will also carry out an initial consolidation of the covered topic on tasks equipped with ready-made drawings. The work promotes the development of attention, observation and logic.

The work consists of the following blocks:

• Types of angles.
• Properties of an inscribed angle.
• Problems whose goal is to find the degree measure of various angles that are inscribed in a circle. They serve for repetition and necessary consolidation of all the material covered.

Category:

Slides:

Information:

• Date of material creation: May 08, 2013
• Slides: 13 slides
• Presentation file creation date: May 08, 2013
• Presentation size: 345 KB
• Presentation file type: .rar
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• Last downloaded: October 15, 2019, at 4:45 pm
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There are 21 presentations in total

Lesson topic: Angles inscribed in a circle. 9th grade.

Lesson objectives:

Educational: get acquainted with the concepts of inscribed and central angles, the inscribed angle theorem and its consequences. Learn to solve problems using the theorem and its consequences. Strengthen the knowledge of low-performing students, strengthen and expand the knowledge of average and well-performing students.

Educational: develop in students the ability to analyze, make comparisons, generalize, build evidence, conduct observations, and plan activities.

Educational: nurturing a culture of mathematical speech; building a response plan; formation of skills to exercise mutual control and self-control.

Equipment:

Multimedia projector

Test (independent work)

Task cards for group work

Cards in orange and blue colors.

During the classes:

Hello, please sit down. Today we have an important, new topic, assignments on this topic are found in the State Academy of Sciences and the Unified State Examination.

What is the name of the topic of the lesson, and what is the purpose of today's lesson, tell me a little later.

And now let's repeat some concepts needed to learn a new topic.

1. What is the name of a segment connecting two points on a circle and passing through the center.

2. How many degrees is a circle? (slide)

3. What figure is called an angle?

4. A triangle whose vertices lie on a circle is called........? (slide)

5. What figure is called an arc of a circle? (slide)

6. Does each corner have......?

We carry out tasks:

Calculate the degree measure of angle ABC.

C angle AOC = 120 0

Student answers. These tasks have become problematic.

Pay attention to how the angle you need to find is constructed. Where is the vertex of the angle?

How are the sides of the angle? What can you call this corner?

Is this a new concept? So the topic of our lesson is......(students' answers)

Let's write down the number and topic of the lesson "Angles inscribed in a circle" (slide)

What is the purpose of our lesson? (students' answers)

Objective of the lesson for students:

Get acquainted with the new concept of inscribed angle; additional concepts related to inscribed angle; learn to calculate the degree measure of an inscribed angle; develop independence.

Construct an inscribed angle and write a definition.

(students' answers) definition slide

Construct an angle whose vertex lies at the center of the circle.

What can you call this corner? (student responses) Make a definition.

Definition slide.

Exercise. Are these angles central or inscribed?

The sides of the central and inscribed angles divide the circle into…….(arcs)

Extend the sides of the corners that you have constructed and use a pen to highlight the arcs located inside the corner.

Do you think an arc has a degree measure? Degree measure, what angle is the degree measure of an arc? (students' answers) Slide

We perform the exercises orally. Find x . slides 5 tasks

(children go to the screen and tell the solution to the problems)

Let's do it now practical task and try to calculate the degree measure of the inscribed angle.

Which figure will the degree measure of the inscribed angle be associated with?

This means that the central and inscribed angles must rest on the same arc.

Complete the constructions and perform the calculations. Draw a conclusion (students’ answers)

Slide.

Let's do the exercise orally.

Slides. 6 tasks

Practical work.

Construct the inscribed angle. Select the arc on which it rests. Construct several more inscribed angles based on this arc. Take measurements and draw a conclusion. (students' answers)

Construct an inscribed angle based on a semicircle. Conclusion (students' answers)

Slide.

Problem solving 7-9 on the slides.

Work in groups.

We do the work individually and check with the students in the group.

Let's check.

Let's repeat the material in the textbook

Let's return to our tasks that we were unable to do at the beginning of the lesson.

Problem solving.

Independent work.

Peer review. Slide.

What did you learn in class today? (students' answers)

If you understand everything today - orange card

If you do not understand all the material - blue card.

Ratings.

Homework: paragraph 107 in 13-16 No. 48(a), 49. Application of inscribed angles in architecture.