Question

5 multiple choice 1 point list the side lengths from shortest to longest. t, r, s s, t, r s, r, t r, s, t

Explanation:

Step1: Find the third angle of the triangle

The sum of angles in a triangle is \(180^{\circ}\). Let the third - angle be \(x\). Using the property of exterior angles (the exterior angle is equal to the sum of the two non - adjacent interior angles), we can also calculate the interior angles.
One interior angle is \(180 - 77=103^{\circ}\), another is \(180 - 71 = 109^{\circ}\), and the third is \(180-(103 + 109)=180 - 212\) (wrong approach). Let's use the correct formula:
If we consider the triangle with angles \(A\), \(B\), \(C\). We know that one angle \(A = 180-103=77^{\circ}\), another angle \(B=180 - 109 = 71^{\circ}\), and the third angle \(C=180-(77 + 71)=32^{\circ}\)

Step2: Apply the angle - side relationship in a triangle

In a triangle, the side opposite the smaller angle is shorter.
Let the side opposite the \(32^{\circ}\) angle be \(t\), the side opposite the \(71^{\circ}\) angle be \(r\), and the side opposite the \(77^{\circ}\) angle be \(s\)
Since \(32^{\circ}<71^{\circ}<77^{\circ}\), by the angle - side relationship (in a triangle, if \(A < B < C\), then \(a < b < c\) where \(a\) is the side opposite angle \(A\), \(b\) is the side opposite angle \(B\), and \(c\) is the side opposite angle \(C\))

Answer:

\(t,r,s\)