Question

which statements regarding \\(\delta efg\\) are true? select three options.

\\(ef + fg > eg\\)

\\(eg + fg > ef\\)

\\(eg - fg < ef\\)

\\(ef - fg > eg\\)

\\(eg + ef < fg\\)

Explanation:

State the Triangle Inequality Theorem

Using the Triangle Geometry knowledge point, for any triangle with side lengths \(a\), \(b\), and \(c\), the sum of the lengths of any two sides must be strictly greater than the length of the remaining side:

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Analyze the first two options

The first option states:

$$ EF + FG > EG $$

This directly matches one of the standard triangle inequalities, so it is true.

The second option states:

$$ EG + FG > EF $$

This also directly matches a standard triangle inequality, so it is true.

Analyze the third option

The third option states:

$$ EG - FG < EF $$

We can rearrange this inequality by adding \(FG\) to both sides:

$$ EG < EF + FG $$

Since the sum of any two sides is greater than the third side, \(EF + FG > EG\) is true, which means \(EG < EF + FG\) is mathematically equivalent and true. Thus, this option is true.

Analyze the remaining options

The fourth option states:

$$ EF - FG > EG \implies EF > EG + FG $$

This contradicts the triangle inequality \(EG + FG > EF\), so it is false.

The fifth option states:

$$ EG + EF < FG $$

This contradicts the triangle inequality \(EG + EF > FG\), so it is false.

Answer:

• (A) EF + FG &gt; EG (Correct answer)
• (B) EG + FG &gt; EF (Correct answer)
• (C) EG - FG &lt; EF (Correct answer)
• (D) EF - FG &gt; EG
• (E) EG + EF &lt; FG