Question

The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is five times the measure of the first angle. The third angle is 16 more than the second. Let x, y, and z represent the measures of the first, second, and third angles, respectively

Answer 1

x = 30 degrees

y = 67 degrees

z = 83 degrees

Explanation:

Let's first set up a few equations. We know:

x + y + z = 180

y + z = 5x

z = y + 16

Let's now solve our equations through substitution. We'll substitute variable z in equations one and two for (y + 16), which is possible thanks to our third equation.

Now, our second equation has become:

y + y + 16 = 5x

So...

5x = 2y + 16

Our first equation is now:

x + y + y + 16 = 180

So...

x + 2y = 180 - 16

And...

x + 2y = 164

Finally...

x = 164 - 2y

Let's take a look at our new first and second equations:

5x = 2y + 16

x = 164 - 2y

Let's substitute again, replacing x in the first equation with (164 - 2y).

Our first equation is now the following:

5(164 - 2y) = 2y + 16

We can apply the Distributive Property to get:

820 - 10y = 2y + 16

So...

-10y - 2y = 16 - 820

-12y = -804

y = 67

Now that we have found y, we can find x and z.

We know that the third angle z is 16 more than the second, y. That was our third equation, z = y + 16.

So, z = 67 + 16.

z = 83.

We know the sum of the y and z is five times the measure of x.

(83 + 67)/5 = 30

To check our work, we can do 67 + 83 + 30 = 180. So we know we are correct.