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The sum of the measures of the angles of a triangle sum to 180 degrees. The sum of the measures of the second and third angles is 3 times the measure of the first angle. The third angle is 13.2 more than the second angle. Find the measure of the third angle.

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theleangreenbean

Systems of Equations

When solving systems of equations, we must first assign variables to given subjects, and reword the problem into equations.

Solving the Question

Let the angles be x, y and z.

We're given:

  • Sum of angles = 180
    ⇒ Rearrange into equation:
    ⇒ [tex]x+y+z=180[/tex]
  • Second angle + third angle = 3 times first angle
    ⇒ Rearrange into equation:
    ⇒ [tex]y+z=3x[/tex]
  • Third angle = 13.2 more than second angle
    ⇒ Rearrange into equation:
    ⇒ [tex]z = 13.2+y[/tex]

Combine the first and second equations using substitution:

[tex]x+y+z=180\\x+3x=180\\4x=180\\x=45[/tex]

Therefore, the first angle measures 45 degrees.

Plug this answer into [tex]x+2y=166.8[/tex] and solve for y:

[tex]x+2y=166.8\\45+2y=166.8\\2y=121.8\\y=60.9[/tex]

Therefore, the second angle measures 60.9 degrees.

Finally, subtract these answers from 180 to find the third angle:

[tex]180-60.9-45\\=74.1[/tex]

Therefore, the third angle measures 74.1 degrees.

Answer

First angle: 45 degrees

Second angle: 60.9 degrees

Third angle: 74.1 degrees