WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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For the next lesson, we are going to start going over some concepts to use for non-right triangles.
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Everything that we went over so far in this unit had to do with right triangles, and now we are going to go over some non-right triangles.
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The first one is Law of Sines: we already went over trigonometric functions (sine, cosine, and tangent); this has to do with sine, the first one.
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The Law of Sines says that, when you have a non-right triangle ABC, with a, b, and c--
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notice how they are lowercase--representing the measures of the sides opposite the angles with measures A, B, and C (capital A, B, C)
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respectively, then this is true, right here; this is actually the Law of Sines.
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Now, let's go over that again: triangle ABC's angles are capital letters--capital A, capital B, capital C.
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Now, here the lowercase a, b, and c are representing the sides opposite those angles.
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For angle A, the side opposite angle A, which is this side right here--we can name this as lowercase a.
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And then, the side opposite angle C is going to be lowercase c, and the side opposite angle B will be lowercase b.
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Whenever you see capital letters, they are going to be representing angles.
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And lowercase letters are going to be representing the sides.
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Since they are the same letters, just keep in mind that it has to be the angle with the side opposite; they have to go together.
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In that case, now, remember again: this is for non-right triangles.
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If we have a right triangle, then we would use the Pythagorean theorem; we would use special right triangles,
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geometric mean, and Soh-cah-toa (trigonometric functions) to find unknown sides and angles--unknown measures.
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This is used for non-right triangles; so if you have a triangle that is not a right triangle, then this can be an option to use.
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Sine of A...remember, you can only find the sine of angle measures...over the side a, equals the sine of B (angle B), over side b;
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that is equal to the sine of capital C, which is the angle C, over side c.
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Now, you are not going to use all three; we are going to use two of the three to make a proportion, and then solve it out that way.
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But it is just letting you know that all three ratios are equal to each other.
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And then, you would just use two of the ratios that you need to find out the unknown value.
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So again, sine of A over a equals sine of B over b, which equals sine of C over c.
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Let's use the Law of Sines to find b, lowercase b; so we know that we are looking for the side opposite angle B; that is this right here.
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Now, the Law of Sines says that the sine of A, over a, is equal to the sine of B, over b; and that is equal to the sine of C, over c.
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So, here I have angle C and side c; I have this, and I have this.
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And then, I have angle B, and I am looking for side b; so I would have to use this one, because here is my unknown value.
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The sine of 85 (B is 85), over b, is equal to the sine of 50 degrees, over a.
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Now, again, only angle measures go with the sine: sine is of angle measures only.
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Then, this would be the side c, which was 8; there we have our proportion.
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Now, we can solve for b, the one variable here.
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Remember: these always have to go together; they can't split up.
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Sine cannot go by itself; it is sine of an angle measure.
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When we cross-multiply, I am going to write b times sin(50).
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Do I multiply anything together? No, I have to keep sin(50) separate.
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It equals...here, 8 times sin(85): again, do you multiply 8 by 85?
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No, you have to keep that separate, because the sine of that angle measure is one number; and then whatever this is--then you can multiply that by 8.
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But you cannot multiply 8 by 85; make sure that you don't do that, because,
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if you multiply 8 by 85, then that no longer becomes the angle measure for sine.
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You can do this two ways: you can solve these out on your calculator, and then solve for b;
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or you can just solve for b first, and then use your calculator.
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Here, take your calculator; you don't have to use a graphing calculator--you can use a scientific calculator--anything with those trigonometric functions.
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The sine of 50--that is the first one: that is .7660, so this right here became .766, and then b is equal to 8 times that.
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We are only solving out the trigonometric functions first: sin(85)...you get .9962.
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Now, from here, you can multiply that by 8, and then divide this.
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So, if I just go ahead and divide this, b is equal to all that; so then, you just use your calculator.
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.9962...I don't have to erase it; I can just leave it on my calculator, and just multiply that by 8; it equals 7.96.
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And then, from there, just go ahead and divide by .766, and you are going to get 10.4, so b is 10.4.
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Again, you see what you have: you need two ratios to make a proportion.
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We have angle C, and we have side c, so we are going to use that.
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And then, b is what we are looking for, so we have to use this one.
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Then, you just plug everything in; you are going to cross-multiply.
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You can solve it out first if you would like; it doesn't matter.
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You can calculate this out and then start cross-multiplying; or you can cross-multiply first, which is what I did, and then you can calculate this and this.
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The reason why I calculated these out first was so that you wouldn't get confused when I started dividing this whole thing.
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But you can divide this whole thing, because you want to solve for b (isolate b), so you can divide the whole thing by sin(50).
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Just make sure that you divide this whole thing together; sin(50) has to stay together--that is one thing that has to stay together.
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And then, it is all calculator from there.
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Moving on: now, here we have what is called solving the triangle.
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When you solve the triangle, you are looking for all unknown values.
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A triangle, we know, has three sides and three angles.
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So, minus what they give you--whatever value is unknown, you need to solve for.
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Here there are going to be six in all, three angles and two sides.
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But here, they give you angle A; they give you side b; and they give you side a.
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Everything that you have to solve for would be the measure of angle C, the measure of angle B, and then...what else is missing?...side AB.
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So, all three measures have to be solved for; they have to be found, and that is called solving the triangle.
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To solve the triangle one thing at a time, what do I have? I have angle A, and I have side a.
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I have side b, and I can find angle B using the Law of Sines.
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Sine of A, over a, equals sine of B, over b, which is equal to sine of C, over c.
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I am going to use this one, and I am going to use this one, because this is my unknown.
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The sine of 45 (that is angle A), over 10, equals the sine of...do I know B?...no, that is what I am looking for, so I leave it like that...over 14.
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So then, I can cross-multiply this; and if you cross-multiply, you are going to get 10sin(B) = 14sin(45).
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Now, again, don't multiply the 14 and the 45 together; do not multiply those numbers together.
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Then, from here, because again, I am solving for B, so I can just divide this 10--get rid of that 10 and divide this whole thing by 10.
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So, the sine of B equals...now, let's use our calculator from here.
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The first thing I am going to do is solve for sin(45); sin(45) is .7071.
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Then, multiply that by 14; you don't have to delete it off of your calculator--just multiply it by 14, and then divide by 10; and you get .9899.
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Now, that is not our answer, because we are looking for this angle measure, B.
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Now, remember that, when we punch this into our calculator...make sure that you don't do sine of this number,
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because then the number you punch in, right after you push sin, is supposed to be the angle measure.
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If you push sin and a number, that number must be the angle measure.
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This is not the angle measure, so we can't punch it in right after sin.
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Since we are looking for the angle measure (we don't have the angle measure; that is what we are looking for), we have to do inverse sine.
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So then, push 2nd and sin; that way, on the top of your sin button, you see sin^-1.
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And then, you are going to punch in .9899, and by doing that, you will get the angle measure, which is 81.8 degrees.
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The measure of angle B is 81.8 degrees; there is B--we found that one.
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Now, to find the measure of angle C, we could do all this again.
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Actually, I wouldn't be able to, because I don't have side c.
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So, if I don't have angle C, and I don't have side c, then I can't do this.
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But what you can do: the best way to do that is to subtract it from 180, because we know that all three angles of a triangle have to add up to 180.
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So then, I am going to do 81.8 degrees, add it to 45, and then subtract that number from 180;
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and then, you get that the measure of angle C is 53.15 degrees.
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I have this, and I have this; and now I need to look for AB, which is side c.
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Now, let me just write c, since that is how it is here: side c.
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Now that I have this (this is 53.15 degrees), I can now go back to this and solve for my side by using this ratio.
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Now, I have everything else, so I can choose this one or this one (it doesn't matter) for my second ratio.
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I am going to just use A; so here it is going to be sin(45)/10 = sin...what is angle C?...51.15, over side c (is what I don't have; that is what I am looking for).
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From here, cross-multiply: csin(45) = 10sin(53.15).
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I can just go ahead and divide this whole thing by sin(45).
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And then, just punch it in your calculator: first, sin(53.15) times 10, divided by sin(45)...and you get (this all canceled out) that side c is 11.317, or 11.3.
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That is this measure right here; so again, from here, just punch in 10 times the sine of that,
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divided by the sine of that, and you should get this answer right here.
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Don't forget parentheses, if you need parentheses.
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That is solving the triangle, just looking for all unknown measures.
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OK, now let's do our examples: Find the measure of angle A here.
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So again, we are going to just use the Law of Sines: sin(A)/a = sin(B)/b = sin(C)/c.
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We don't have this; we have a; and then, we have the c's, angle C and side c.
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I am going to use this one and this one: sine of angle A, over 16, equals sine of 62, over 21.
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21sin(A) = 16sin(62); again, don't multiply these together.
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Divide the 21; sin(A) = 16sin(62), divided by 21; that is .6727.
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Now, I am looking for angle A; that means I have to do inverse sine, because it is not angle measure that I have.
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So, I have to use inverse sine, 2nd and sin: .6727...the measure of angle A is 42.28 degrees.
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And that is all we had to find, the measure of angle A.
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Example 2: Here, they don't give you a triangle; they are just giving you these measures.
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a is 10; the measure of angle B is 40; the measure of angle A is 55 degrees; find b.
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We know that we are going to use sin(A)/a = sin(B)/b.
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So, sine of...what is A?...the measure of angle A is 55 degrees, over 10,
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equals the sine of...the measure of angle B is 40, over b (is what we are looking for).
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To continue here: bsin(55) = 10sin(40); divide sin(55); don't separate these--sin and 55 have to go together.
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Divide this by the sine of 55 degrees; you just take your calculator: 10 times the sine of 40, divided by the sine of 55: b = 7.8.
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All of this b equals 7.8; just make sure that you multiply the sine of 40...punch in sin(40), multiply that by 10, and divide it by sin(55).
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And the next one: here is side c, side b, angle B...and find angle C.
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I am going to use sin(B) over side b equals sine of angle C, over side c.
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The sine of B is 42, over 15, is equal to the sine of C (that is what we are looking for), over...side c is 12.
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Cross-multiply: 15sin(C) = 12sin(42); divide the 15; sin(C) = 12sin(42)/15, so it is 0.5353.
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I need to find my angle measure, so I am going to push 2nd and then sin, the inverse sine, of .5353.
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The measure of angle C is 32.36 degrees.
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We are going to solve the triangle; that means that we have to look for the measure of angle C; we have to look for side c and side a.
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To find the measure of angle C, since we are given two angles--whenever you are given two angles, you just have to subtract them from 180.
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52 plus 48 becomes 100; so 180 - 100 is 80 degrees--that is the measure of angle C.
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And then, let's look for a and c; to find a, I'll do...let's just go straight into the Law of Sines.
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The sine of angle A is 52, over side a, equals...we can't use this yet, so it would be sin(48)/9.5.
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asin(48) = 9.5sin(52); divide the sine of 48; so a is going to be...find 9.5sin(52); and we divide this by sin(48); and that is 10.07.
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a is going to be 10.07; now, let's look for side c.
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We are going to do sine of AD, sine of C, over c, equals...well, let's use B; so sin(48) over 9.5.
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c times the sine of 48 (cross-multiplying) equals 9.5 times sin(80).
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Divide the sine of 48; c equals (it is all a calculator from here) 9.5 times sin(80), divided by sin(48); so I am going to get 12.6 here for c.
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One way to check to see if your answers are correct: this is 80; this is 12.6; and this was 10.1 or 10.07;
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angle B has the smallest measure--that means that side b should be the shortest side.
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Next is angle A: the measure of angle A is 52, so it should be slightly bigger than side b, which is 10.07.
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And then, angle C has the largest angle measure; this is the largest angle--that means that the side opposite it has to be the largest side.
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So then, smallest to biggest: this side should be the smallest, and this side should be the biggest, depending on how big the angle measures are.
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If we solve for c, and we get 20, or if we get 2, then we know that that is wrong, because,
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since angle C has the largest angle measure, the side opposite has to be the largest side; it has to be the longest measure.
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So, if it is smaller than any of these numbers, then you know that there is something wrong; you did something wrong.
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It is just one way to check.
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Your last one: Find the length of BC and BD.
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Here is BC, and here is BD; so then, here I know that this is a parallelogram, because we have two pairs of opposite sides being parallel.
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These two are parallel, and these two are parallel; and that is by definition of parallelogram that this is parallelogram ABCD.
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Let's see, now: we know that we are on triangles, so we are going to be using something with these triangles here.
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They want us to find BC (so then here, let's say this is x) and BD (let's say this is y).
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Now, for this one here, let's see: here I know that this is going to be 8, because this is the side opposite,
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and then, because these are congruent (in a parallelogram, opposite sides have to be congruent), what else do I know?
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I know that this angle and this angle are congruent, because they are alternate interior angles (parallel lines...transversal...so alternate interior angles).
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The same thing here: because these two lines are parallel, and this is a transversal, and this is 72, then this has to be 72.
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And then, you can use those two: subtract from 180, and then that will be angle C; but I don't think we will need that.
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Here, we can use the Law of Sines now to find this side: an angle with the opposite and an angle with this opposite.
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The sine of 50, over 8, is going to equal the sine of angle measure 72, over x.
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Cross-multiply: xsin(50) = 8sin(72); divide the sine of 50; so x is going to be 8 times the sine of 72, divided by sin(50).
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So, I am going to get 9.9 for this length right here; and that is going to be BC--that was 9.9.
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So then, that one is done; and then, to find BD, let's see; I do need that angle measure, because that is the angle opposite that side.
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So here, this is going to be...50 + 72 is 122; 180 - 122 is going to be 58 degrees.
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This is 58; to find the y, I can use any pair, so let's just use this one, because this is a whole number.
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The sine of angle measure 50, over the side opposite, 8, equals the sine of 58, over y.
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y times sin(50) equals 8 times sin(58); divide sin(50); y equals...and that becomes 8.86, or 8.9; and that will be BD; it is 8.9.
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Now, why is it that, if we have this side here, and we have x (we have BC right there, 9.9)--
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why can't we use the Pythagorean theorem to find this third side?
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That is because this is not a right triangle; you can only use the Pythagorean theorem for right triangles.
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This is the Law of Sines; you can only use it for non-right triangles, any time that you have a triangle where they all have to be acute angles.
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They are acute angles; no right triangles; and you are looking for angle measures or side measures.
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Then, you are going to be using the Law of Sines.
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That is it for this lesson; thank you for watching Educator.com.