QuestionThe sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is three times the measure of the first a...

Using the fact that the sum of the measures of the three angles of a triangle is $180^{\circ} .$ In a triangle, the measure of the first angle is three times the measure of the second angle. The measure of the third angle is $35^{\circ}$ less than the measure of the second angle. What is the measure of each angle?

So here again, we have a triangle. How you're told that you're told that one of the angles let's just say this one. Let's call it a and it's our angle. Let's call it be mutual that one of the angles A is, uh, five more, then the retains the other angle. The really be So this is the aggression, Debbie. How so? If that's the case, then uh, we know that a is five plus the repeat that also retort that the last single, this one, is equal to 115 decrees and so disagree Now. Now we need to find the angles right. You need to find it to unknown angles. To do that. I mean, it is the fact that some of the angles in a triangle as 0 to 180 180 equals 115 us A. Plus B. Now it is five plus three b. So I can just say that vehicles to run Shafik Scene plus five plus three B plus B or 180 is equal to 120. I'm adding 5 to 115 plus for being adding three B two B now the subtract one Tony from here and also from here to get 16 equals two for e Nice 60 is for B. Unless the white both sides were for Do I decide by for? And though I decide before and we get 15 he calls to be 15 then Now, if 15 Ys b then the other angle A is fine, plus three times speak or three times 15 on Daddy is the retired 15 is 45. It's gonna become 50 degrees, so today is 50 degrees and be his 15 degrees. So our measurements are these two numbers.

In this course, and we have given lingers angle one that hurt England. You have to find them. Deserves Thank is Oh, I think this gangland you and well is that Oh, herding, baby one. Okay, eggs. Plus it's is Thanks as you want. Some of them is Inglis angle, is it? According X? Less eggs on it. Now on combining light else and then be writing sites six X is working in full visitors being Inglis. Oh, are thinking are you begin?

Our given caution is one angle of a triangle measures twice the second angle, So let this be any triangle. Let this be a triangle, ABC Now it is given there one angle ever tangle measures twice. The second angle so late and will be is twice twice angle. See similarly, the part angle midges three times. The second angle that is angle is three times angle B. Now we have to find the measure of the angles that is angle E and will be, and then we'll see. As we know that angle a less and will be plus angle see will be called 1 90 degree because these are angles of a triangle. I will be MEES by single see and angle is Christ and will be so put will put everyone in terms of Bengal sea so late angle is three times and will be so three and will be plus and will be. Plus angle C is equal to 1 90 degree, so it becomes four times and will be. Plus angle. C is equal to 1 90 degree, which gives, but I will be. We can replace it with two time single see so according toa air times angle C plus and they'll see physical toe. Wonder treated me so it gives nine times angle cuz equal to one artistically, which gives angle. See musical to 20 degrees. Now if NDC is 20 degrees so and will be used twice. Angle see, so it becomes two times 20 degrees. That is 40 degree and angle is just for three times angle B. So it becomes three multiplied very 40 that is 1 20 degrees. So angle A it's 1 20 degree and will be this 40 diddly and angle sees 20 TV, that is, this is 20 degree, this is 40 degree and this is 1 20

In this triangle with angle measures eight degrees be degrees and C degrees. We have to find a B and C so that the second angle B is 10 degrees greater than the first, and the third angle is the same size as the other two combined to solve. For these three variables A, B and C, we need a system of three equations, and so here we have two so far that were given in the question or whose information where is given in the question. And we conform the third equation by considering the triangle and remembering that the total the sum of the angles in a triangle is always 180 degrees. And so now we have our system. How can we start solving this system? Well, we want to form either just an equation one equation in one variable or first, a system of two equations in two variables and then use that to form one equation with one variable. However, we don't need to do that since we have a plus B here, which we can just replace with C. Since we know they're the same thing we cannot replace, not April's be we can replace the April's be in Equation three with C, and we'll get any question with just the variable seat. And so let's do that now. Here is the equation with just see, and it tells us that C equals 90 degrees. And now we can use that value of C to substituted back into another equation if we substituted into either Equation two or Equation three, we find in both cases that April's B equals 90. And so now we have two equations with just A and B a system of two equations into variables. And then we can solve that, using familiar methods, substitution and elimination. Let's substitute again, since we have be isolated in Equation one and then let's simple by that what we're and we find a equals 20. No, we can find the value would be using Equation one, and from there, refined B equals 70. Now that we have the values of A B and C, well, you can double check that we got them right by substituting them back into the original statement of the question and see if they make sense