Question

The sum of the measure of the angles in triangle ABC is 180 degrees. The sum of the measures of angle B and angle C is twice the measure of angle A. The measure of angle B is 32 degrees less than the measure of angle C.

Answer 1

`a=60\\b=44\\c=76`

Explanation:

You can write these equations to represent the problem:

`a+b+c=180\\b+c=2a\\b=c-32`

Now, just solve it as a system of equations. I'll solve by substitution, starting from the bottom as b is already defined there.

`b=c-32\\b+c=2a\\(c-32)+c=2a\\c+c-32=2a\\2c-32=2a\\a=c-16`

That's the middle equation done, now I'll do the top one:

`b=c-32\\a+b+c=180\\a+(c-32)+c=180\\a+2c-32=180\\a+2c=212\\a=-2c+212`

Now we have just 2 equations with 2 variables. That's another system of equations, but this one is much easier to solve. Using substitution again:

`a=c-16\\a=-2c+212\\c-16=-2c+212\\3c-16=212\\3c=228\\c=76`

Finally, we have one variable solved. Using the value of C, now we can solve for A:

`c=76\\a=c-16\\a=76-16\\a=60`

Thats 2 done. Now, pick any equation that has all 3 variables and solve for B:

`a=60\\c=76\\a+b+c=180\\60+b+76=180\\b+136=180\\b=44`

All 3 variables done.

`a=60\\b=44\\c=76`

Now, we can confirm that they're correct by checking with the original equations:

`a+b+c=180\\60+44+76=180\\180=180\\\\b+c=2a\\44+76=2(60)\\120=120\\\\b=c-32\\44=76-32\\44=44`

They all work fine.