While measuring the angles in a triangle, Patricia found the measure of one of the angles is \(45\degree\). Once she had measured the other two angles, she noticed that the measures of these two angles are in the ratio \(4 : 5\). What is the measure of each of the other two angles?
Solution 1
Since the measure of the unknown angles are in the ratio of \(4:5\), then we can give the angles measures of \(5n\degree\) and \(4n\degree\).
Since the sum of the measures of the three angles in any triangle is \(180\degree\), then \[\begin{aligned} 45 + 4n + 5n &= 180\\ 4n + 5n &= 135\\ 9n &= 135\\ n &= 15 \end{aligned}\] Therefore, the other two angles are \(5n = 5(15) = 75\degree\) and \(4n = 4(15) = 60\degree\).
Solution 2
Since the sum of the measures of the three angles in any triangle is \(180\degree\), then the sum of the measures of the two unknown angles in the triangle is \(180\degree - 45\degree= 135\degree\).
The measures of the two unknown angles are in the ratio \(4 : 5\), and so one of the two angles measures \(\frac{5}{4+5} = \frac{5}{9}\) of the sum of the two angles, while the other angle measures \(\frac{4}{4+5} = \frac{4}{9}\) of the sum of the two angles.
That is, the larger of the two unknown angles measures \(\frac{5}{9} \times 135\degree = 75\degree\), and the smaller of the unknown angles measures \(\frac{4}{9} \times 135\degree = 60\degree\).
Therefore, the other two angles are \(75\degree\) and \(60\degree\).