contestada

A school theater department is trying to set up their stage pulley system
for a play, but they need a triangular block with specific angle
measurements for one of the set pieces. The second angle of the triangle
has a measure equal to the sum of two times the other two angles. The third
angle has a measure equal to ten less than the first angle. What are the measures of all three angles?

Respuesta :

Using a system of equations, it is found that:

  • The measure of the first angle is of 35º.
  • The measure of the second angle is of 120º.
  • The measure of the third angle is of 25º.

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  • We are going to call the angles x, y and z.

In a triangle, the sum of the interior angles is of 180º, thus:

[tex]x + y + z = 180[/tex]

The second angle of the triangle  has a measure equal to the sum of two times the other two angles, then:

[tex]y = 2(x + z)[/tex]

[tex]x + z = 0.5y[/tex]

Replacing into the first equation, we find the second angle:

[tex]x + y + z = 180[/tex]

[tex]0.5y + y = 180[/tex]

[tex]1.5y = 180[/tex]

[tex]y = \frac{180}{1.5}[/tex]

[tex]y = 120[/tex]

The measure of the second angle is of 120º.

Now, working to find the measures of the first and of the third angles:

[tex]x + z = 0.5y \rightarrow x + z = 60 \rightarrow x = 60 - z[/tex]

The third  angle has a measure equal to ten less than the first angle, then:

[tex]z = x - 10[/tex]

Since [tex]x = 60 - z[/tex]

[tex]z = 60 - z - 10[/tex]

[tex]2z = 50[/tex]

[tex]z = \frac{50}{2}[/tex]

[tex]z = 25[/tex]

And

[tex]x = 60 - z = 60 - 25 = 35[/tex]

The measure of the first angle is of 35º.

The measure of the third angle is of 25º.

A similar problem is given at https://brainly.com/question/13387714