Question

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

Explanation:

Step1: Recall the theorem

In a triangle, the side opposite the larger angle is the larger side.

Step2: Analyze the first triangle

In the first triangle, the angles are $30^{\circ}$, $50^{\circ}$, and $100^{\circ}$. The side opposite the $100^{\circ}$ angle has length 13, the side opposite the $50^{\circ}$ angle has length 10, and the side opposite the $30^{\circ}$ angle has length 5. Since $100^{\circ}>50^{\circ}>30^{\circ}$ and $13 > 10>5$, it follows the theorem.

Step3: Analyze the second triangle

In the second equilateral triangle, all angles are $60^{\circ}$ and all sides are 5. There is no larger - angle - larger - side relationship here as all angles and sides are equal.

Step4: Analyze the third triangle

In the third triangle, the angles are $23^{\circ}$, $67^{\circ}$, and $90^{\circ}$. The side opposite the $90^{\circ}$ angle (hypotenuse) should be the longest. But here the side opposite the $67^{\circ}$ angle has length 13 and the side opposite the $90^{\circ}$ angle has length 12, which does not follow the theorem.

Answer:

The first triangle.