How do I find the angles of an isosceles triangle whose two base angles are equal and whose third angle is 10 less than three times a base angle?

Since you're looking for the measurement of the angles, you can begin this problem by assigning a variable to each angle. So let's call the two base angles a and b and the third angle c. Because the sum of the angles of a triangle equals 180, you know that
a + b + c = 180

You also know that the two base angles are the same, which means that a = b. So you can rewrite this equation as

a + a + c = 180 or 2a + c = 180

You know that the third angle (c) is "10 less than 3 times a base angle" (which in this case is a). This can be written mathematically as

c = 3a – 10

Now substitute for c in the equation 2a + c = 180 and you can solve for a:

2a + 3a – 10 = 180 (group the a's together and add 10 to both sides of the equation)
5a = 190 (divide both sides by 5)
a = 38 (which also means that b = 38; you've solved for two of the three angles)

Now substitute for a in c = 3a – 10 and solve the equation:

c = 3(38) – 10
c = 114 – 10
c = 104

And there you have it. The three angles measure 38 degrees, 38 degrees, and 104 degrees. To check your answer, figure out whether these three angles add up to 180 degrees like they should.