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Question

One angle in a triangle has a measure that is three times as large as the smallest angle. The measure of the third angle is 50 degrees more than that of the smallest angle. Find the measure of the LARGEST angle.

Answer 1

Hi!

I think the best way to solve this would be to set up a system of equations. Since there are 3 unknowns (the 3 angles), we will need a system of (at least) 3 equations to determine the unknown values. Let's see if we can come up with 3 equations using known facts.

#1: One angle is three times as large as the smallest angle. Let's call these angles y and x, with x being the smallest one in the triangle. Here's an equation that reflects this statement:

y = 3x

#2: The third angle is 50 degrees larger than the smallest angle. Let's call this third angle z. The smallest angle is x, as we are still referring to the same triangle as before. Equation:

z = x + 50

#3: We can take advantage of the fact that the sum of a triangle's angles is 180 degrees. Equation:

x + y + z = 180

So, our system of equations is as follows:

y = 3x

z = x + 50

x + y + z = 180

We can solve this equation using substitution. Since the first two equations give us expressions for the values of y and z, we can substitute these expressions into the last equation in place of y and z. The last equation will then look like this:

x + y + z = 180

x + (3x) + (x + 50) = 180

We can then solve for x by combining the terms containing x into one value, and subtracting 50 from both sides:

x + 3x + x + 50 = 180

5x = 180 - 50

5x = 130

x = 26

Now we know that the smallest angle, x, measures 26 degrees. To find the measure of the largest angle, we can determine the values of y and z and see which one is larger. We can use the first two equations from our system of equations and substitute the value of x (which is 26) wherever there is an x:

y = 3x = 3*26 = 78

z = x + 50 = 26 + 50 = 76

So y=78 and z=76. Thus, the largest angle measures 78 degrees. To quickly check our work, we can add up the three values and make sure that their sum is 180. Indeed, 26+78+76 equals 180.

Hope this helps!

~Kaleab