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  • Hi and welcome to Math Antics.

  • In our last geometry video, we learned some important things about angles.

  • One of the things we learned was than angles come in different sizes.

  • Some are big and some are small.

  • Well in this video, we're gonna learn how we can tell exactly how big or small an angle is.

  • We're gonna learn how angles are measured.

  • You probably already know a lot about measurement,

  • like you know how to measure how long something is with a ruler or a tape measure.

  • And the units you'd use would be inches, or centimeters, or something like that, right?

  • But when it comes to angles, we can't use a ruler to measure them, or use units like centimeters.

  • And that's because angles aren't about length, angles are about rotation.

  • And to measure how much something is rotated, we use a special unit called degrees.

  • Now hold on a second, I thought degrees were used to measure how hot or cold something is.

  • Ya know, like, "it's 100 degrees outside today!"

  • Ah, that's a good point you smart looking fellow. The word degree is actually used for

  • a lot of different things, so it can be a little confusing sometimes.

  • It makes more sense if you just think of a degree as a small amount of something.

  • For temperature, a degree is a small amount of heat.

  • But for angles, a degree is a small amount of rotation.

  • And there's a special math symbol for degrees

  • that we can use instead of writing the word 'degrees' over an over again.

  • It's this little circle that you put after the number and up near the top.

  • To see how we use degrees to measure angles,

  • let's get two rays that point in exactly the same direction.

  • Then, let's put one ray directly on top of the other one, so it looks

  • like there is only one ray there, even though there's really two.

  • Now, let's take the ray on top and rotate it just a tiny amount counter-clockwise.

  • This point on the ray will be our axis (or center) of rotation.

  • It is like the point at the center of a clock that stays stationary while the hands rotate around it.

  • Our rays now form an angle that measures 1 degree, and as you can see, 1 degree is a really small angle.

  • We need to zoom in on it to see that it really is an angle.

  • In fact, you might wonder if there could be any angle smaller than 1 degree.

  • Yep, there sure are. And we saw one just a second ago.

  • Before we rotated our top ray, when our rays were exactly on top of each other,

  • that is a zero degree angle.

  • And there's a whole range of tiny fraction angles between 0 and 1 degree,

  • but we aren't going to learn about them in this video.

  • Instead, we are going to keep on rotating our top ray and watch the angle get bigger and bigger.

  • This special readout here will tell us how many degrees our angle measures.

  • Now let's start out slow. 1 degree, 2, 3, 4, 5, 6, 7, 8, 9, and 10, now let's hold it there for a second.

  • So this is what 10 degrees looks like.

  • 10 degrees?! That's f-f-freezing!

  • Huh. Guess you're not as smart as I thought after all.

  • So we can see that a 10 degree angle is still a very small angle.

  • So let's keep going, but a little bit faster this time.

  • That's 15 degrees, 20, 25, 30, 35, 40 and 45.

  • Now 45 degrees is a special angle because it's exactly half of a right angle.

  • If we draw a right angle in the same spot, you can see that our ray cuts it into two equal parts.

  • So, if 45 is half of a right angle, can you guess how many degrees a right angle is?

  • Let's keep on rotating to see if you're right. 50, 60, 70, 80, and 90.

  • Yep, a right angle is exactly 90 degrees,

  • and that is super important to memorize because right angles are used all the time in geometry.

  • Okay, do you remember from our last video

  • that all angles less than a right angle are called acute angles?

  • So that means that all the angles we've seen so far that are between

  • 0 and 90 degrees (like 10, 30, 45, 60 and so on) are acute angles.

  • But, if we keep on rotating our ray past 90 degrees,

  • we'll start forming obtuse angles, because they are greater than a right angle.

  • Ready? Here we go. 100 degrees, 110, 120, 130, 140, 150, 160, 170 and 180.

  • Ah ha, does this look familiar? Yep. It's a straight angle, like we learned about in the last video.

  • The rays point in exactly opposite directions

  • and the angle they form is 180 degrees.

  • And that's also a really important angle measurement to memorize.

  • Now before we go on, let's quickly review the important angles and regions we've looked so far.

  • Our angle measurement is zero degrees when the rays point in the same direction.

  • It's 90 degrees when they are perpendicular, and form a right angle.

  • And it's 180 degrees when they point in opposite directions, and form a straight angle.

  • In this region (between 90 and 180) we find obtuse angles.

  • And in this region (between 0 and 90) we find acute angles.

  • One important acute angle is 45 degrees since it's half of a right angle.

  • Alright then, let's continue rotating past 180 degrees.

  • Our angle readout keeps getting higher, and the next important angle we come to is this one, 270 degrees.

  • It also forms a right angle, but it points down instead of up.

  • Let's keep on going because another really important angle is just around the corner.

  • And it's coming up right about now!

  • We've rotated our ray all the way around the axis and now it's back to where we started.

  • Now you might be wondering, "If we're back where we started, then why is our counter

  • reading 360 degrees instead of 0 degrees like before?"

  • The answer is that, even though our rays are back to the same place,

  • we had to rotate our top ray 360 degree to get it there.

  • And you can see that our angle arc now forms a complete circle.

  • So 360 degrees is the angle that represents a full circle! Rotating 360 degrees brings

  • you all the way around the circle to the point where you started from.

  • Okay, now that you know what degrees are and have seen how they relate to the size of an angle,

  • we need to learn how to actually measure an angle without this fancy readout that we have here.

  • Just like a ruler can be used to measure the length of a line,

  • a special tool called a protractor can be used to measure angles.

  • A protractor is similar to a ruler, but it's curved into a half-circle so that it can measure

  • rotation around an axis point. A protractor also has a straight edge with a hole or dot

  • in the middle that represents the axis, or center of rotation.

  • So, if you are given a mystery angle (like this one)

  • and you want to measure how many degrees it is,

  • you just put your protractor on top of it

  • so that the axis point lines up with the intersection of your rays, like this.

  • Then you make sure that one of the rays lines up with the straight line on the protractor.

  • And last of all, you look to see where the other ray crosses the curved part

  • and read off what angle measurement it lines up with.

  • As you can see, this angle is 50 degrees.

  • Alright, there's one more thing I want to teach you in this video because you will probably

  • see this kind of geometry problem on your homework or tests.

  • Do you remember what complementary and supplementary angles are from the last video?

  • Complementary angles combine to form a right angle,

  • and supplementary angles combine to form a straight angle.

  • Well, now that we know a right angle is 90 degrees, and a straight angle is 180 degrees,

  • we can use that information to solve problems that have unknown angles, like this one.

  • It shows two angles (A & B) that combine to form a right angle.

  • The problem tells us that angle A is 30 degrees, and it wants us to figure out what angle B is.

  • Fortunately, it's easy to figure that out now

  • because we know that a right angle is 90 degrees,

  • so we know what the total of both angles must be.

  • That means that to find angle B, all we do is take the total (which is 90 degrees)

  • and subtract angle A (Which is 30 degrees) and whatever is left over will be angle B.

  • So, 90 - 30 = 60. Angle B is 60 degrees.

  • Now let's try this problem. It uses the same idea, but with a straight angle this time.

  • The straight angle is divided into two smaller angles. (again, angle A & angle B)

  • And again, the problem tells us that angle A is 70 degrees, and it wants us to figure out what angle B is.

  • Well, we know that the total of both angles must be 180 degrees

  • because we just learned that's how big a straight angle is.

  • So if we take that total (180 degrees) and subtract

  • angle A (which is 70 degrees), whatever is left after subtracting must be angle B.

  • So 180 - 70 = 110.

  • Pretty cool, huh? And now you can see why it's important to know how degrees work in geometry.

  • They can tell us how big angles are or how much something is rotated.

  • Well, that's all I've got for you in this video.

  • But don't worry, there's a lot more geometry where that came from.

  • So I'll get going on my next video, and you get going on practicing what you've learned.

  • Thanks for watching Math Antics, and I'll see ya next time.

  • Learn more at mathantics.com.

Hi and welcome to Math Antics.

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数学のアンチティクス - 角度と度数 (Math Antics - Angles & Degrees)

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    Yassion Liu に公開 2021 年 01 月 14 日
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