Question

applying the triangle parts relationship theorem
which triangle correctly shows that the side opposite the larger angle is the larger side?

Explanation:

Step1: Analyze the first triangle

In the first triangle, the angles are \(90^{\circ}\), \(67^{\circ}\), \(23^{\circ}\). The side opposite \(90^{\circ}\) is \(13\), opposite \(67^{\circ}\) is \(12\), opposite \(23^{\circ}\) is \(5\). But \(12>5\) while \(67^{\circ}>23^{\circ}\) and \(13\) is the hypotenuse (longest side as it's opposite the right - angle). However, the side opposite \(67^{\circ}\) (\(12\)) should be longer than the side opposite \(23^{\circ}\) (\(5\)), which is correct, but let's check others.

Step2: Analyze the second triangle

In the second triangle, all angles are \(60^{\circ}\). By the property of an equilateral triangle (all angles equal, all sides equal). But the sides are given as \(6\), \(8\), \(8\), which is incorrect for an equilateral triangle. So it doesn't show the side - angle relationship (since angles are equal but sides are not all equal as they should be for equal angles).

Step3: Analyze the third triangle

In the third triangle, angles are \(100^{\circ}\), \(50^{\circ}\), \(30^{\circ}\). The side opposite \(100^{\circ}\) is \(13\), opposite \(50^{\circ}\) is \(10\), opposite \(30^{\circ}\) is \(6\). Since \(100^{\circ}>50^{\circ}>30^{\circ}\) and \(13 > 10>6\), the side opposite the larger angle is the larger side.

Answer:

The third triangle.