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 22 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.

x=30°, y=64°, z=86°

Step-by-step explanation:

Make a system of equations using the information given.

x + y + z = 180    <-- from "the sum of interior angles equals 180 degrees"

y + z = 5x           <-- from "sum second angle and third angles is five times the first angle"

z = y + 22           <-- from "the third angle is 22 more than the second"

Eliminating the x variable to makes the equations easier to solve.

5x + 5y + 5z = 900

-(5x                = z + y)

=       5y + 5z = 900 - z - y

= 6y + 6z = 900

= y + z = 150

Now substitute z for (y+22)

y + (y + 22) = 150

2y + 22 = 150

2y = 128

y = 64

Now find z when y=64

z = (64) + 22

z = 86

Finally, plug in z and y into one of the equations with x and evaluate for x.

x + 64 + 86 = 180

x + 150 = 180

x = 30

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