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Rom4ik [11]
3 years ago
14

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 three time

s the measure of the first angle. The third angle is 29 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.
Mathematics
1 answer:
Rzqust [24]3 years ago
8 0
<h3>Answers:</h3>
  • x = 45
  • y = 53
  • z = 82

====================================================

Explanation:

The three angles add to 180, so x+y+z = 180

The second and third angles add to 3 times the measure of the first, so y+z = 3x

The third angle is 29 more than the second, so z = y+29

----------------------------

The system of equations is

\begin{cases}x+y+z = 180\\\\y+z = 3x\\\\z = y+29\end{cases}

For now, let's focus on the second and third equations only.

Start with y+z = 3x. Replace z with y+29 which is valid because of the third equation.

So we'll have

y+z = 3x

y+y+29 = 3x

2y+29 = 3x

Let's isolate y

2y+29 = 3x

2y = 3x-29

y = (3x-29)/2

y = 3x/2 - 29/2

y = 1.5x - 14.5

We'll use this later.

--------------------------

Go back first equation of the system. Plug in z = y+29

x+y+z = 180

x+y+y+29 = 180

x+2y+29 = 180

Then plug in y = 1.5x - 14.5 we found earlier

x+2y+29 = 180

x+2(1.5x-14.5)+29 = 180

x+3x-29+29 = 180

4x = 180

x = 180/4

x = 45

---------------------------

We can use this x value to find y

y = 1.5x - 14.5

y = 1.5*45 - 14.5

y = 53

and use this to find z

z = y+29

z = 53+29

z = 82

As a check, x+y+z = 45+53+82 = 180 to help confirm the answer. I'll let you check the other equations.

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