Question

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, angles are \(30^\circ\), \(50^\circ\), \(100^\circ\) and sides are \(5\), \(10\), \(13\). Check the side opposite the largest angle (\(100^\circ\)): the side opposite \(100^\circ\) is \(10\), the side opposite \(50^\circ\) is \(13\), and the side opposite \(30^\circ\) is \(5\). Here, the side opposite the larger angle (\(50^\circ>30^\circ\)) is \(13\) which is larger than \(5\), but the side opposite \(100^\circ\) (largest angle) is \(10\) which is smaller than \(13\) (opposite \(50^\circ\)), so it does not follow the rule.

Step2: Analyze the second triangle

In the second triangle (right - angled), angles are \(90^\circ\) (right angle), \(67^\circ\), \(23^\circ\) and sides are \(5\), \(12\), \(13\). The largest angle is \(90^\circ\), the side opposite to it is \(13\) (hypotenuse). The angle \(67^\circ\) is larger than \(23^\circ\), the side opposite \(67^\circ\) is \(12\) and the side opposite \(23^\circ\) is \(5\). We have \(90^\circ>67^\circ>23^\circ\) and the opposite sides are \(13 > 12>5\), which follows the rule that the side opposite the larger angle is the larger side.

Step3: Analyze the third triangle

In the third triangle, all angles are \(60^\circ\) (equilateral triangle), so all sides are equal (\(5 = 5=5\)). Since all angles are equal, all sides are equal, so it does not show the relationship between larger angles and larger sides (as there are no different - sized angles to test the rule).

Answer:

The Middle Triangle (the second triangle in the given set of three triangles)