Question

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 30 more than the second. Let T. y, and z represent the measures of
the first, second, and third angles, respectively. Find the measures of the three angles.
Provide your answer below:

Answer 1

• t = 30°
• y = 60°
• z = 90°

Explanation:

The given relations can be used to write three equations for the angle values.

Setup

t + y + z = 180 . . . . . sum of all angle measures

y + z = 5t . . . . . . . . second and third total 5 times the first

y + 30 = z . . . . . . third angle is 30 more than the second.

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Solution

Using the third equation to substitute for z in the first two equations gives ...

t + y + (y +30) = 180 ⇒ t +2y +30 = 180

y + (y +30) = 5t ⇒ 2y +30 = 5t

Using the second of these equations to substitute for (2y+30) in the first, we have ...

t +5t = 180

t = 180/6 = 30 . . . . . . divide by the coefficient of t

Using this value in our equation for 5t gives ...

2y +30 = 5(30)

2y = 120

y = 60

Then the value of angle z is ...

y +30 = z = 60 +30

z = 90

The first, second, and third angles are 30°, 60°, and 90°, respectively.