1. Suppose that in a certain triangle, the degree measures of the interior angles are in the ratio 2:3:4.If the largest interior angle measures x degrees, what is the value of \(x^\circ\)?

2.Suppose that in a certain triangle, the degree measures of the exterior angles are in the ratio 2:3:4.

If the largest interior angle measures x degrees, what is the value of\(x\circ\) ?( x degrees)

3.

Guest Aug 1, 2017

#1**+1 **

1. Suppose that in a certain triangle, the degree measures of the interior angles are in the ratio 2:3:4.If the largest interior angle measures x degrees, what is the value of ?

I'm assuming that you want the value of the largest angle....the sum of the interior angles of a triangle = 180° ... and the ratio of 2:3:4 indicates that there there are 9 equal parts of the180°......the largest angle is 4/9ths of this = (4/9) (180) = 80°

2.Suppose that in a certain triangle, the degree measures of the exterior angles are in the ratio 2:3:4.

If the largest interior angle measures x degrees, what is the value of ?( x degrees)

The sum of the exteriior angles of any regular polygon total 360°.......as before, there are 9 equal parts of the 360°....the largest interior angle will be supplementary to the * smallest* exterior angle.......the smallest exterior angle = (2/9) (360) = 80°....so the greatest interior angle will = 180 - 80 = 100°

Proof of this is that the next greatest exteriior angle = (3/9)(360) = 120° and the interior angle supplementary to this = 180 - 120 = 60°

And the greatest exterior angle = ( 4/9) (360) = 160°....and the interior angle supplementary to this = 180 - 160 = 20°

And the sum of the interior angles is 100 + 60 + 20 = 180°

Thanks to the guest for spotting my previous error.....!!!!!

CPhill
Aug 1, 2017

#3**0 **

Here's a method that you can utilize

If the ratio of the triangle's angle is 2:3:4, that means that the there is an angle with the value 2y, 3y, and 4y. Of course, by the triangle sum theorem, all the angles in a triangle must equal 180 degrees. Now, let's solve for y:

\(2y+3y+4y=180\) | Simplify the left hand side of the equation. 2y+3y+4y=9y. |

\(9y=180\) | Divide by 9 on both sides of the equation. |

\(y=20\) | |

To find the measure of the largest angle, substitute the value for x and plug it into the largest angle, which is 4x since 4 multiplied by a natural number is greater than one multiplied by three:

\(4(20)\) | |

\(y=4*20=80^{\circ}\) | |

2)

This problem is very similar to the one above. This time, however, we must know that the sum of all the exterior angles in any polygon is equal to 360 degrees. We'll use y again and solve just like before:

\(2y+3y+4y=360\) | Combine like terms on the left hand side of the equation. |

\(9y=360\) | Divide by 9 on both sides. |

\(y=40\) | |

Ok, we want the smallest exterior angle to get the largest interior angle:

\(180-(2*40)\) | Evaluate this |

\(180-80\) | |

\(100\) | |

Therefore, the largest interior angle is 100 degrees.

TheXSquaredFactor
Aug 1, 2017