Find the supplement of the smallest angle of the triangle

Question

One angle of the triangle is 50. If the remaining angles are in the ratio of 4 to 9, find the supplement of the smallest angle of the triangle.

Answer 1

A triangle's interior angles will always add up to 180. Let's say the unknown angles are x and y, and we can make an equation by adding up all the sides like below:

x + y + 50 = 180

Now we have two unknown variables, which means we need at least two independent equations. We have our first, let's figure out how to make a second. We know that the ratio of the two unknown sides is 4/9, so we could say:

x/y = 4/9

Now that we have our two equations, we just need to write one equation in terms of one variable. To do this, we need to figure out what x is equal to in terms of y. Solve for x in the second equation:

x/y = 4/9

x = 4/9 * y

= 4y/9

Now go back to the first equation, and replace x with what is equal to in terms of y to have an equation with only one variable:

x + y + 50 = 180

4y/9 + y + 50 = 180

Now we can solve for y:

4y/9 + y + 50 = 180

4y/9 + y = 130

4y + 9y = 1170, here I just multiplied everything by 9 to get rid of the fraction

13y = 1170

y = 90

Now we know y is 90, so we can plug this into the equation we wrote for x to solve for it:

x = 4y/9

= 4(90)/9

= 40

To check our answer, we can plug both these into the original equation:

x + y + 50 = 180

40 + 90 + 50 = 180, so it looks like everything worked.