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in \\( \triangle \mathrm { def } \\), \\( \mathrm { ef } = 19 \\), \\( \mathrm { fd } = 7 \\), and \\( \mathrm { de } = 17 \\). which
statement about the angles of \\( \triangle \mathrm { def } \\) must be true?
answer
\\( \mathrm { m } \angle f < \mathrm { m } \angle e < \mathrm { m } \angle d \\)
\\( \mathrm { m } \angle e < \mathrm { m } \angle d < \mathrm { m } \angle f \\)
\\( \mathrm { m } \angle d < \mathrm { m } \angle f < \mathrm { m } \angle e \\)
\\( \mathrm { m } \angle f < \mathrm { m } \angle d < \mathrm { m } \angle e \\)
\\( \mathrm { m } \angle e < \mathrm { m } \angle f < \mathrm { m } \angle d \\)
\\( \mathrm { m } \angle d < \mathrm { m } \angle e < \mathrm { m } \angle f \\)

Explanation:

Step1: Recall the triangle angle - side relationship

In a triangle, the larger the length of a side, the larger the measure of the angle opposite that side. That is, if in $\triangle ABC$, $BC = a$, $AC = b$, $AB = c$, then the larger the value of $a$, the larger the measure of $\angle A$ (the angle opposite side $a$), and so on.

Step2: Identify the sides and their opposite angles in $\triangle DEF$

• Side $EF = 19$: The angle opposite to $EF$ is $\angle D$ (because in $\triangle DEF$, side $EF$ is opposite vertex $D$).
• Side $FD = 7$: The angle opposite to $FD$ is $\angle E$ (because side $FD$ is opposite vertex $E$).
• Side $DE = 17$: The angle opposite to $DE$ is $\angle F$ (because side $DE$ is opposite vertex $F$).

Step3: Compare the lengths of the sides

We are given that $EF = 19$, $FD = 7$, and $DE = 17$. Comparing the lengths: $FD=7

Step4: Relate the side lengths to the angle measures

Since the angle opposite a shorter side is smaller, we have:

• The angle opposite $FD$ (which is $\angle E$) is smaller than the angle opposite $DE$ (which is $\angle F$), so $m\angle E
• The angle opposite $DE$ (which is $\angle F$) is smaller than the angle opposite $EF$ (which is $\angle D$), so $m\angle F

Combining these two inequalities, we get $m\angle E

Wait, side $FD = 7$ (opposite $\angle E$), side $DE = 17$ (opposite $\angle F$), side $EF = 19$ (opposite $\angle D$). So:

Since $FD = 7

Wait, let's re - identify the opposite angles correctly:

• In $\triangle DEF$, the angle opposite side $DE$: side $DE$ is between $D$ and $E$, so the angle opposite $DE$ is $\angle F$.
• The angle opposite side $EF$: side $EF$ is between $E$ and $F$, so the angle opposite $EF$ is $\angle D$.
• The angle opposite side $FD$: side $FD$ is between $F$ and $D$, so the angle opposite $FD$ is $\angle E$.

So side lengths: $FD = 7$, $DE = 17$, $EF = 19$. So $FD

Wait, let's list the sides with their opposite angles again:

• Side $DE$: connects $D$ and $E$, so opposite angle is $\angle F$.
• Side $EF$: connects $E$ and $F$, so opposite angle is $\angle D$.
• Side $FD$: connects $F$ and $D$, so opposite angle is $\angle E$.

Given $EF = 19$, $FD = 7$, $DE = 17$. So the side lengths in order: $FD = 7$, $DE = 17$, $EF = 19$. So the angles opposite: $\angle E$ (opposite $FD$), $\angle F$ (opposite $DE$), $\angle D$ (opposite $EF$). So $\angle E<\angle F<\angle D$? But that's not an option. Wait, maybe I made a mistake in the side - angle correspondence.

Wait, another way: Let's label the triangle properly. Let's denote the triangle with vertices $D$, $E$, $F$. Then:

• Side opposite $\angle D$: $EF$ (because $\angle D$ is at vertex $D$, so the side opposite is $EF$).
• Side opposite $\angle E$: $FD$ (because $\angle E$ is at vertex $E$, so the side opposite is $FD$).
• Side opposite $\angle F$: $DE$ (because $\angle F$ is at vertex $F$, so the side opposite is $DE$).

So side lengths: $EF = 19$ (opposite $\angle D$), $FD = 7$ (opposite $\angle E$), $DE = 17$ (opposite $\angle F$).

Now, compare the side lengths: $FD = 7$, $DE = 17$, $EF = 19$. So $FD

Answer:

1. $m\angle E < m\angle F < m\angle D$