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. Find the measures of the three angles.

Answer 1

Explanation:

Let x be the measure of the first angle.

According to the problem, we know that:

The sum of the angles of the triangle is 180: x + y + z = 180

The sum of the second and third angles is five times the measure of the first angle: y + z = 5x

The third angle is 16 more than the second: z = y + 16

We can substitute the third equation into the second equation to get:

y + (y + 16) = 5x

Simplifying this equation, we get:

2y + 16 = 5x

We can rearrange this equation to get:

y = (5/2)x - 8

Now we can substitute this equation and the equation z = y + 16 into the first equation to get:

x + (5/2)x - 8 + (5/2)x + 8 = 180

Simplifying this equation, we get:

6x = 360

Dividing both sides by 6, we get:

x = 60

Now we can use this value of x to find y and z:

y = (5/2)x - 8 = (5/2)(60) - 8 = 58

z = y + 16 = 58 + 16 = 74

Therefore, the measures of the three angles are x = 60, y = 58, and z = 74.