Question

which triangles area can be calculated using the trigonometric area formula?

Explanation:

Step1: Recall trigonometric area formula

The trigonometric area formula for a triangle is $A=\frac{1}{2}ab\sin C$, where $a$ and $b$ are two - side lengths of the triangle and $C$ is the included angle between them.

Step2: Analyze each option

For the first triangle (triangle with vertices A, B, C), only side - lengths are given, no angle. So, we can't use the trigonometric area formula.
For the second triangle (triangle with vertices D, E, F), we have two side - lengths (20) and two angles ($100^{\circ}$ and $34^{\circ}$). We can use the trigonometric area formula $A = \frac{1}{2}ab\sin C$. Here, we can take the two given sides and the included angle information to calculate the area.
For the third triangle (triangle with vertices H), there is not enough information (only one side - length and one angle) to use the trigonometric area formula.

Answer:

The triangle with vertices D, E, F.