Final answer:
Using the Law of Sines and considering the Triangle Inequality Theorem, we find that the only possible value for ∞F in ΔDEF, to the nearest degree, is approximately 84°.
Step-by-step explanation:
To find all possible values of ∞F in triangle DEF given f = 210 cm, e = 310 cm, and ∞E = 25°, we can use the Law of Sines, which states:
(sin E)/e = (sin F)/f
First, we calculate sin E which is sin(25°). Then we solve for sin F:
sin F = (f/e) * sin E
sin F = (210 cm / 310 cm) * sin(25°)
We then find the value of sin F and use the inverse sine function to find the measure of angle F, paying attention to the fact that there might be two solutions in the range of 0° to 180° because the sine function is positive in the first and second quadrants. However, we must also consider the Triangle Inequality Theorem which states that the sum of any two sides of a triangle must be greater than the measurement of the third side. In our case, since side DEF is larger than 210 cm, angle F will only have one solution because the sum of angles E and F must be less than 180°.
Finally, we round the measure of angle F to the nearest degree.
After calculating, you will find that the only possible value for ∞F to the nearest degree, that satisfies the conditions of the Law of Sines and Triangle Inequality Theorem, is approximately 84°.