Solve each problem. See Examples 5 and 9. The sum of the measures...

Question

Solve each problem. See Examples 5 and 9. The sum of the measures of the angles of any triangle is 180°. In a certain triangle, the largest angle measures 55° less than twice the medium angle, and the smallest angle measures 25° less than the medium angle. Find the measures of all three angles.

Accepted Answer

Hello, everybody. I hope we're doing all right. Today, today, we're going to be looking at this math question that states in a triangle ABC, the measure of the smallest angle is 20 degrees less than the measure of the medium angle. And the measure of the largest angle is 50 degrees less than thrice the measure of the medium angle. And they're asking us to find all three angles of the triangle. Now, as your choice, A is 100 degrees, 50 degrees and 30 degrees. B is 100 and 10 degrees, 60 degrees and 10 degrees C is 100 and 20 degrees, 40 degrees and 20 degrees N D is 90 degrees, 60 degrees and 30 degrees. Now, when looking at this question, we have three different angles and different values for each one of them. So I, one of the first things we wanna do is assign a variable to each one of the angles just so it's a little bit more organized. So we'll have that will have largest angle. I'm going to assign it to variable X medium angle. I will assign it to variable Y and smallest angle. I will assign it to variable Z. Now with these variables and using the information that they gave us, we can write two different equations for two of the angles. So they tell us the smallest angle is 20 degrees less than the medium angle. So the smallest angle is Z is equal to the medium angle minus 20 degrees. This is just how to write what they told us. But with numbers, so they told us 20 degrees less than the medium angle. So the medium angle minus 20 degrees and that is equal to the smallest angle. And then they also told us that the largest angle. So in this case, X is 50 degrees less than thrice the mesh, the measure of the medium angle. So thrice the ma thrice the measure of the medium angle is three times or three multiplying the measure of the medium mango. So that's three Y and then they also said it's 50 degrees less than that. So it would be three Y minus 50 degrees. So the measure of the largest angle X is equal to three Y minus 50 degrees. Now, we know because this is a triangle, we know that the measure of the largest angle X plus plus the measure of the medium angle Y plus the measure of the smallest angle Z will be equal to 100 and 80 degrees because this is a triangle. So now that we have these three formulas, we can do some plugging in so I can plug in the formula we obtain for Z and the formula we obtain sir for X into the X plus Y plus Z formula. So that's exactly what we'll do. So for X, we'll plug in three Y minus 50 degrees and then that would be plus Y still. And now, instead of having that plus Z, we'll have plus Y minus 20 degrees, which is the formula we obtain for Z and that'll still be equal to 100 and degrees. Now, we can do a bit of simplifying and I'm just going to move my work a bit to the left. So I have some more room to work with and I'll rewrite our equation. So we'll have three Y minus 50 degrees plus Y plus Y minus 20 degrees equals 100 and 80 degrees. Now, like I mentioned, we can do some simplification now. So we have a three Y and we have plus Y and plus Y. So that's two Y. So it'll be three Y plus two Y which is five Y and then we had a minus 50 degrees or a negative 50 degrees and a minus degrees or same thing, same as saying negative 20 degrees. So we had a negative 50 degrees minus 20 degrees, which is minus 70 degrees. So we had five Y now we have five Y minus 70 degrees equals 100 and 80 degrees. Then we can move the 70 degrees over from the left hand side to the right hand side by adding it. So we'll have five, Y is equal to 250 degrees. Which means that once we divide each side by five, we have that Y is equal to 50 degrees. So that is one of the answers for our angle. This will be the medium angle. Now, we can plug this in to our equations. R Z and our X equations. And then we'll do X first. So we had that X is equal to three Y minus 50 degrees. So now by plugging that Y, we have that X is equal to three multiplying degrees, which was the value we obtained for Y minus 50 degrees. So X is equal to three multiplied by 50 which is 100 and 50 degrees minus 50 degrees. And this means that X is equal to 150 minus 50 which is 100 degrees. And this will be our second answer for an angle. In this case, X, which is the largest angle. And finally, we can plug our Y value into our other equation for the smallest angle. So we have that Z is equal to Y minus 20 degrees, which means that Z is equal to R Y which was 50. So we have 50 minus 20 degrees, which means that Z is equal to 30 degrees. And that is the answer for our smallest angle. And now we have obtained all of the angles. So our largest angle will be 100 degrees. Our medium angle will be 50 and our smallest angle will be 30 which matches up with answer choice. A so I hope that was helpful. And until next time.

Solve each problem. See Examples 5 and 9. The sum of the measures of the angles of any triangle is 180°. In a certain triangle, the largest angle measures 55° less than twice the medium angle, and the smallest angle measures 25° less than the medium angle. Find the measures of all three angles.

Key Concepts

Triangle Angle Sum Theorem

Algebraic Expressions

System of Equations

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