measuring, constructing, and using angles
DESCRIPTION
Measuring, constructing, and using angles . Contents Why Measure Angles? How might we measure angles? What is an “angle”? What does “measurement” mean? Our “measuring tool” for angles. Contents (continued) Measuring an angle Drawing an angle of a given size - PowerPoint PPT PresentationTRANSCRIPT
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Measuring, constructing,
and using angles
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ContentsWhy Measure Angles?How might we measure angles?
What is an “angle”?What does “measurement” mean?Our “measuring tool” for angles.
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Contents (continued)Measuring an angle Drawing an angle of a given sizeThe “radian” unit of measure for angles.
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Why measure angles?
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Why measure angles?
So we can figure outhow big the Earth is!
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If we choose two places that are north and south of each other,
A
B
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then measure the distance between them,
Dist
ance
A
B
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and also the highest angle the Sun reaches at each place on a given day,
Highest angle at Location A Highest angle at Location B
Sky simulations made with the free astronomy program Stellarium (http://stellarium.org/)
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we can calculate the size of the Earth.
The first person to do this calculation was Eratosthenes, a librarian at the Great Library of Alexandria, Egypt, around 200 BC.
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Actually, almost all of astronomy and geography depends upon being able to measure angles.
So do many jobs, such as surveying.
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So, how might we measure angles?
First, let’s review what we mean by “angle”, and “measurement”.
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One definition of “angle” is
“A pair of rays thathave the same endpoint”.The endpoint, V, is called the vertex.
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You’ve probably learned that angles are related to “turns”, and that the same set of rays can be made by two different turns.
For example, here are …
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the two “turns” that take us from ray VA to ray VB.
Counterclockwise turn Clockwise turn
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Now that we’ve reviewed what “angle” means, what do we mean by “measurement”?
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A good definition is found in The Archimedes Codex: How a Medieval Prayer Book is Revealing the True Genius of Antiquity’s Greatest Scientist:*
* Reviel Netz and William Noel, Da Capo Press, 2009, p. 41.
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“To measure is to find a measuring tool and apply it successively to the object being measured. Suppose we want to measure a straight line.
According to the authors,
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“For instance, suppose we want to measure your height, which is really saying that we want to measure the straight line from the floor to the top of your head.
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“Then what we do is take a line the length of an inch [this is our measuring tool] and apply it successively, well over sixty times, but probably fewer than eighty times to measure your height.
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“Since this is very tiresome, we have pre-marked measuring tapes that save us the trouble of actually applying the [one-inch line] successively,
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“but, at the conceptual level, successive application is precisely what takes place.”
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That definition of “measurement” needs some explanation. For example, when we read
“To measure is to find a measuring tool and apply it successively to the object being measured,”
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we probably thought of “measuring tools” as rulers, etc.
Actually, the authors meant something quite different.
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They meant that to measure a length (that’s the example they give), the “tool” we choose is a line segment with a convenient length.
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The authors mentioned “an inch” as an example of a convenient length, but we could use a segment of any length we like.
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For example, we could use this yellow segment:
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Now, suppose that we wanted to “measure” this red segment.
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We’d start at one end of the red segment, and “apply” the yellow segment to it repeatedly, until we got to the other end …
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1
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2
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3
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Here’s the summary of what we just did to “measure” the red segment.
1
2
3
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Because we had to apply the yellow segment three times,
1
2
3
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we say that the length of the red segment is “3 yellow segments”.
1
2
3
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For example, if our yellow segment were an inch, the length of the red one would be “three inches”.
1
2
3
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As the authors of The Archimedes Codex told us, this process would be tiresome if we had do many measurements.
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For that reason, we make pre-marked rulers and measuring tapes,
1 2 3 4 5
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so that we may “apply” several “yellows” at once.
1 2 3 4 5
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If we wished, we could also mark off fractional parts of our yellow segment, just as we do with inches on a ruler.
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Now that we’ve reviewed what “angle” and “measuring” mean,
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let’s learn the common ways of measuring an angle.
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As we’ve seen, the “measuring tool” that we use for lengths is some convenient segment.
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However, when we measure angles, the “measuring tool” that we use is some sector of a circle.
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For example, this one:
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To measure an angle,
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we align one side of our sector with one of the angle’s rays,
Note: “Point” of the sector must be aligned with vertex of the angle.
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Note: “Point” of the sector must be aligned with vertex of the angle.
then keep “applying” our sector until we reach the other ray.
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Note: “Point” of the sector must be aligned with vertex of the angle.
then keep “applying” our sector until we reach the other ray.
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We use the same method for the other direction of “turn”:
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Note: “Point” of the sector must be aligned with vertex of the angle.
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Note: “Point” of the sector must be aligned with vertex of the angle.
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Note: “Point” of the sector must be aligned with vertex of the angle.
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Note: “Point” of the sector must be aligned with vertex of the angle.
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Note: “Point” of the sector must be aligned with vertex of the angle.
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Note: “Point” of the sector must be aligned with vertex of the angle.
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Note: “Point” of the sector must be aligned with vertex of the angle.
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Note: “Point” of the sector must be aligned with vertex of the angle.
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Note: “Point” of the sector must be aligned with vertex of the angle.
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Note: “Point” of the sector must be aligned with vertex of the angle.
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Note: “Point” of the sector must be aligned with vertex of the angle.
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Note: “Point” of the sector must be aligned with vertex of the angle.
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Note: “Point” of the sector must be aligned with vertex of the angle.
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Note: “Point” of the sector must be aligned with vertex of the angle.
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Note: “Point” of the sector must be aligned with vertex of the angle.
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Although we could choose any sector as our “measuring tool”, the most-common is the one we get when we divide a circle into 360 equal sectors:
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The angle enclosed by each of those sectors is called one degree.
We abbreviate that as “ 1° ”.
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Important:No matter how big our circle,
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a “ 1° ” sector always contains the same angle.
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So, now we have our 1° measuring tool.
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Let’s use it to measure an angle.
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We had to apply our “ 1° ” tool eleven times,
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so we say that the angle “measures 11°”.
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Now that we have our angle-measurement tool, and know how to use it, we can look for ways to make a more-convenient version of it.
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We already know that we can make a more-convenient version of a length-measurement tool by pre-marking “something” at chosen intervals, to make a tape or ruler.
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For measuring angles, we can pre-mark 1° sectors on something:
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However, this tool is “messy”, so we “clean it up” by …
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leaving marks only at the center and edges of our circle.
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Those are all we need for measuring an angle.
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To make a “nicer” tool, we can number the degrees, and add a few other extras, to make the tool we call …
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a protractor …
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which comes in several versions.
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To measure an angle with a protractor,
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we align a “zero” line of the protractor with a ray of the angle …
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we align a “zero” line of the protractor with a ray of the angle …
Note where protractor aligns with vertex.
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we align a “zero” line of the protractor with a ray of the angle …
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we align a “zero” line of the protractor with a ray of the angle …
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we align a “zero” line of the protractor with a ray of the angle …
Note where protractor aligns with vertex.
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then read how many degrees we’d “apply” to get to the other ray.
90°
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Notice that we’ve measured the shorter of this angle’s two “turns”.
90°
![Page 101: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/101.jpg)
Here’s the longer “turn” for this angle.
![Page 102: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/102.jpg)
How many degrees should it measure?
![Page 103: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/103.jpg)
We’ll discuss that later.
![Page 104: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/104.jpg)
The 90° angle (also called a right angle) is an important one to know.
![Page 105: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/105.jpg)
So is the angle that measures 180°. It’s called a “straight angle”.
![Page 106: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/106.jpg)
So is the angle that measures 180°. It’s called a “straight angle”.
180°180°
![Page 107: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/107.jpg)
So is the angle that measures 180°. It’s called a “straight angle”.
![Page 108: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/108.jpg)
So is the angle that measures 180°. It’s called a “straight angle”.
180°180°
![Page 109: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/109.jpg)
90° 180°
Knowing these two angles…
![Page 110: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/110.jpg)
helps us when we have to measure an angle like this one.
![Page 111: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/111.jpg)
Does it measure 60°, or 120°?
![Page 112: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/112.jpg)
By counting, we can see that there are 60 degrees between the rays.
60°
![Page 113: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/113.jpg)
Also, 60° minus 0° is 60°. (That’s what we read on the inner scale).
60°
![Page 114: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/114.jpg)
180° - 120° is 60°, too. (That’s what we read on the outer scale).
60°
![Page 115: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/115.jpg)
Our angle 90°
Finally, our angle (the one we want to measure) is smaller than 90°.
![Page 116: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/116.jpg)
All of these observations show that our angle measures 60°, not 120°.
60°
![Page 117: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/117.jpg)
For comparison, here’s an angle that does measure 120°.
![Page 118: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/118.jpg)
For comparison, here’s an angle that does measure 120°.
120°
![Page 119: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/119.jpg)
Our angle 90°
We can see that it’s bigger than 90°,
![Page 120: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/120.jpg)
Our angle 180°
but smaller than 180°.
![Page 121: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/121.jpg)
Please remember that when we say that these angles measure 60° and 120°,
![Page 122: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/122.jpg)
we’re talking about the measure of the smaller of their “turns”,
![Page 123: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/123.jpg)
not the larger one.
![Page 124: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/124.jpg)
These larger “turns” measure more than 180°, but less than the 360° that there are in a full circle.
![Page 125: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/125.jpg)
They’re examples of what are called “reflex angles”.
![Page 126: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/126.jpg)
So is the “larger turn” of a 90° angle, which we saw earlier.
![Page 127: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/127.jpg)
We’ll now look, briefly, at how to find the measure of a reflex angle.
![Page 128: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/128.jpg)
First, we can break the reflex angle into a straight angle (pink turn), …
![Page 129: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/129.jpg)
and a 60° angle (orange turn).
![Page 130: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/130.jpg)
The measure of the reflex angle is the sum of these: 180° + 60° = 240°.
60°
180°
![Page 131: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/131.jpg)
240°
![Page 132: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/132.jpg)
A second way is to remember that in a full circle, there are 360°.
![Page 133: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/133.jpg)
Therefore, the number of degrees in the reflex angle has to be …
![Page 134: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/134.jpg)
360° minus 120° (= 240°).
360°120°
![Page 135: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/135.jpg)
240°
![Page 136: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/136.jpg)
A third way to measure a reflex angle …
![Page 137: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/137.jpg)
is with a full-circle protractor.
240°
![Page 138: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/138.jpg)
Here are links to more information about reflex angles:
Reflex angleshttp://www.mathsisfun.com/reflex.html
Re-entrant angleshttp://www.thefreedictionary.com/re-entrant+angle
![Page 139: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/139.jpg)
Besides wanting to measure angles, there are times when we want to (or must!) draw them.
![Page 140: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/140.jpg)
Someone will tell us (or we will choose) the following things:
![Page 141: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/141.jpg)
Where the vertex is,
![Page 142: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/142.jpg)
the ray from which the angle is to start,
![Page 143: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/143.jpg)
the direction of the “turn”, and
![Page 144: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/144.jpg)
the measure of the angle that we are to draw.
105°
![Page 145: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/145.jpg)
Now, just as we did for measuring angles, …
105°
![Page 146: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/146.jpg)
we align a protractor with the vertex and ray.
105°
![Page 147: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/147.jpg)
Next, we go in the direction of the given turn,
105°
![Page 148: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/148.jpg)
for the required number of degrees.
105°
Here’s the mark for 105°.
![Page 149: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/149.jpg)
We make a little dot there on our paper, …
105°
Here’s the mark for 105°.
![Page 150: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/150.jpg)
then remove our protractor, ...
105°
![Page 151: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/151.jpg)
and draw a ray from the vertex through our dot.
105°
![Page 152: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/152.jpg)
We’ve now constructed the required angle.
105°
![Page 153: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/153.jpg)
We’ll finish this discussion of angles by mentioning another common “tool” for measuring them.
![Page 154: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/154.jpg)
We imagine making that tool by drawing an circle, and one of its radii, …
![Page 155: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/155.jpg)
then making a segment as long as the radius, …
![Page 156: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/156.jpg)
sticking one end of the segment to the circle, …
![Page 157: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/157.jpg)
and bending the segment to fit the circle.
![Page 158: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/158.jpg)
Now, we draw rays from the center of the circle through the endpoints of the segment.
![Page 159: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/159.jpg)
The angle we’ve “cut out” of the circle is defined as “1 radian”.
![Page 160: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/160.jpg)
This “measuring tool” has a property that makes it useful in subjects that you’ll study later.
![Page 161: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/161.jpg)
That property is that
In a full circle, there are 2π radians.
![Page 162: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/162.jpg)
Can you see why this is so?
Hint: What is the relationship between the length of a circle’s radius, and the length of its circumference?
![Page 163: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/163.jpg)
Summary
Rather than review the details of how to measure angles, I’d like to emphasize that …
![Page 164: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/164.jpg)
Summary (cont’d)
• Many types of work depend upon being able to measure angles. For example, astronomy and surveying.
![Page 165: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/165.jpg)
Summary (cont’d)
• There are different ways of measuring angles.
![Page 166: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/166.jpg)
Summary (cont’d)
• When we need to find a way to measure something, we can invent our own “tools”.
![Page 167: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/167.jpg)
Summary (cont’d)
• It’s often helpful to imagine some simple version of a tool, or of solving any problem, then look for a way of making that simple tool or technique more convenient.
![Page 168: Measuring, constructing, and using angles](https://reader035.vdocuments.mx/reader035/viewer/2022062410/56816221550346895dd24937/html5/thumbnails/168.jpg)
END