Question

angle side relationship
question
in \\( \triangle efg, fg = 11, ge = 15 \\), and \\( ef = 18 \\). which list
has the angles of \\( \triangle efg \\) in order from smallest to
largest?
answer
\\( m\angle g, m\angle e, m\angle f \\)
\\( m\angle f, m\angle e, m\angle g \\)
\\( m\angle e, m\angle f, m\angle g \\)
\\( m\angle e, m\angle g, m\angle f \\)
\\( m\angle g, m\angle f, m\angle e \\)
\\( m\angle f, m\angle g, m\angle e \\)

Explanation:

Step1: Recall the Angle-Side Relationship

In a triangle, the larger the side, the larger the angle opposite to it. So we first identify the sides and their opposite angles.
- Side \( FG = 11 \), opposite angle \( \angle E \) (since in \( \triangle EFG \), side \( FG \) is opposite \( \angle E \))
- Side \( GE = 15 \), opposite angle \( \angle F \) (side \( GE \) is opposite \( \angle F \))
- Side \( EF = 18 \), opposite angle \( \angle G \) (side \( EF \) is opposite \( \angle G \))

Step2: Order the Sides by Length

Order the sides from smallest to largest: \( FG = 11 \), \( GE = 15 \), \( EF = 18 \), so \( FG < GE < EF \).

Step3: Order the Angles by Their Opposite Sides

Since the angle opposite the smaller side is smaller, the angles opposite these sides will follow the same order.
- Opposite \( FG \) (smallest side) is \( \angle E \), so \( \angle E \) is the smallest angle.
- Opposite \( GE \) (middle side) is \( \angle F \), so \( \angle F \) is the middle angle.
- Opposite \( EF \) (largest side) is \( \angle G \), so \( \angle G \) is the largest angle. Wait, no, wait. Wait, let's correct: Wait, side \( FG = 11 \) (opposite \( \angle E \)), \( GE = 15 \) (opposite \( \angle F \)), \( EF = 18 \) (opposite \( \angle G \)). So sides: \( 11 < 15 < 18 \) (FG < GE < EF). Therefore, angles opposite: \( \angle E < \angle F < \angle G \)? Wait, no, that can't be. Wait, no, wait: Wait, in triangle \( EFG \), vertices are E, F, G. So side opposite E is FG, side opposite F is GE, side opposite G is EF. So:

- Side opposite \( \angle E \): \( FG = 11 \)
- Side opposite \( \angle F \): \( GE = 15 \)
- Side opposite \( \angle G \): \( EF = 18 \)

So sides: \( 11 (FG) < 15 (GE) < 18 (EF) \). Therefore, angles: \( \angle E < \angle F < \angle G \)? Wait, no, that would mean \( m\angle E, m\angle F, m\angle G \)? But wait, let's check the options. Wait, maybe I mixed up the opposite angles. Wait, let's label the triangle: \( \triangle EFG \), so the sides are:

- \( FG \): between F and G, so opposite vertex E.
- \( GE \): between G and E, so opposite vertex F.
- \( EF \): between E and F, so opposite vertex G.

Yes, so:

- \( FG \) (length 11) opposite \( \angle E \)
- \( GE \) (length 15) opposite \( \angle F \)
- \( EF \) (length 18) opposite \( \angle G \)

So the order of sides from smallest to largest: \( FG (11) < GE (15) < EF (18) \). Therefore, the angles opposite them (from smallest to largest) are \( \angle E \) (opposite FG), \( \angle F \) (opposite GE), \( \angle G \) (opposite EF). Wait, but that would be \( m\angle E, m\angle F, m\angle G \), but let's check the options. Wait, the options include \( m\angle E, m\angle F, m\angle G \) as one of the choices? Wait, looking at the options:

Options:

1. \( m\angle G, m\angle E, m\angle F \)

2. \( m\angle F, m\angle E, m\angle G \)

3. \( m\angle E, m\angle F, m\angle G \)

4. \( m\angle E, m\angle G, m\angle F \)

5. \( m\angle G, m\angle F, m\angle E \)

6. \( m\angle F, m\angle G, m\angle E \)

Wait, wait, I think I made a mistake. Wait, let's re-express:

Wait, side \( FG = 11 \): connects F and G, so opposite angle is \( \angle E \) (vertex E).

Side \( GE = 15 \): connects G and E, so opposite angle is \( \angle F \) (vertex F).

Side \( EF = 18 \): connects E and F, so opposite angle is \( \angle G \) (vertex G).

So sides: \( FG = 11 \), \( GE = 15 \), \( EF = 18 \). So \( FG < GE < EF \). Therefore, angles opposite: \( \angle E < \angle F < \angle G \)? Wait, no, that would mean \( m\angle E < m\angle F < m\angle G \), so the order from smallest to largest is \( m\angle E, m\angle F, m\angle G \), which is option 3: \( m\angle E, m\angle F, m\angle G \). Wait, but let's check again. Wait, maybe I had the opposite. Wait, no: in a triangle, longer side opposite larger angle. So if side EF is the longest (18), then angle opposite EF (which is angle G) is the largest. Side GE is 15 (opposite angle F), so angle F is middle. Side FG is 11 (opposite angle E), so angle E is smallest. So angles: E (smallest), F (middle), G (largest). So order from smallest to largest: \( m\angle E, m\angle F, m\angle G \), which is option 3.

Wait, but let's check the options again. The third option is \( m\angle E, m\angle F, m\angle G \). Yes. So that's the correct order.

Answer:

\( \text{m}\angle E, \text{m}\angle F, \text{m}\angle G \) (the third option: \( \boldsymbol{\text{m}\angle E, \text{m}\angle F, \text{m}\angle G} \))