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Math 2 | Phase 2 Week 4 Days 1/2 | Central & Inscribed Angles

Autor:Todd Fogdall

Lesson Overview & Quick Angle Review

This week you will learn about two specific types of angles and their relationship to a circle. Today, we will just define the two types of angles and learn the relationship between their measures. (You should have already downloaded this week's notes so that you can fill them in as you go through this activity.) To encourage you to interact with these concepts more than just passively watching a video, we will play a couple games of "warm, hot, on fire." Before we start, though, we should review some basic ideas about angles. Below you'll see a sketch of an angle. Play around with the angle to remind yourself of what makes an angle acute, obtuse, or right. (It's hard to make it a right angle, so if you can get pretty close, I'll give you a button to do the last little bit.)

Question #1 - About Questions in GeoGebra

(NOTE: The questions in this lesson do not confirm whether you are right or wrong. When you show the answer, you are only confirming whether you are right for your own knowledge. In other words, you're not penalized for wrong answers.) QUESTION: When you click to show the correct answer, does GeoGebra confirm whether you're right or wrong? (Yes, this is a "multiple" choice question with only one "choice".)

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Question #2 - Obtuse Angles

What makes an angle obtuse?

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Question #3 - Acute Angles

What makes an angle acute?

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Question #4 - Right Angles

What makes an angle right?

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Question #5 - Naming Angles

In the sketch, I labeled the angle "". There is one other choice of labels that I could have used. Write that choice in the space below, and then check to see if you're right. (In the notes, I ask you a question to see if this review is all making sense.)

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NEW Concept #1 - Defining Central Angles

The first of the two types of angles you'll learn about today is called a "central angle". As I said earlier, we'll play a game of "warm, hot, on fire" to learn the definition of a central angle. In the sketch below, you'll see a circle and an angle with its vertex inside the circle. Play around with the sketch until you get a message that says "Getting warmer." and then "You're hot!". At that point, you're close to making a central angle, and a button will appear to give you a little help to make it a central angle. Click the button.

Question #6 - Define Central Angle

What makes an angle a central angle?

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Questions #7 - Measuring the Central Angle

Once you have a central angle, grab point S and move it around. What do you notice about how this angle is measured that is different from what you may be used to?

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Question #8 - Measuring a Central Angle's Intercepted Arc

You should notice that as soon as you made a central angle, part of the circle turned blue and thicker than the rest of the circle. This portion of the circle is called an "arc", and you should fill in your notes before answering question 8. Definition - Central angles intercept a portion of a circle, which we call an "arc". (A "minor arc" is less than half of a circle. A "major arc" is more than half of a circle. A "semicircle" is exactly half of a circle.) What simple relationship do you notice about the measure of the arc and the measure of the central angle?

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NEW Concept #2 - Defining and Finding Measures of Inscribed Angles

The second of the two types of angles you'll learn about today is called an "inscribed angle". As I said earlier, we'll play a game of "warm, hot, on fire" to learn the definition of an inscribed angle. In the sketch below, you'll see a circle and an angle with its vertex inside the circle. Play around with the sketch until you get a message that says "Getting warmer." and then "You're hot!". At that point, you're close to making an inscribed angle, and a button will appear to give you a little help to make it inscribed. Click the button.

Question #9 - Define Inscribed Angle

What makes an angle an inscribed angle?

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Be careful with this definition!

Not all angles with vertices on a circle are inscribed. The angle's sides must also intersect the circle at exactly two points. In other words, in the picture below, is an inscribed angle because it intersects the circle at points D and B. , however, is not inscribed, because it only intersects the circle at point E. (We'll see in a bit why this matters.)

One inscribed angle, and one non-inscribed angle.

One inscribed angle, and one non-inscribed angle.

Question #10 - Measuring Inscribed Angles

As soon as you made inscribed in the sketch above, you should notice that an arc appears from point E to point S, as well as a central angle that intercepts the same arc. Now move points S or E around, but keep H between them. Observe how big the measure of is (shorthand: ) compared to the central angle, that intercepts the same arc. This isn't quite as obvious as the relationship you saw in Question #8, but I still think you'll see how big the inscribed angle is. (Take a guess, and then check to see if you're right by checking your answer below.) How big is the inscribed angle's measure compared to the central angle's measure?

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Conclusion

That’s it for Days 1 & 2. Before you start on the practice exercises, try the practice problems on your notes page. Then check your work at the video below to be sure that you’re understanding! Good luck!

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