Question

which side is the longest? triangle fge with angles at f: 55°, at g: 63°, at e: 62° options: ge (red), lmnop (blue), ef (orange), fg (green)

Explanation:

Step1: Recall triangle side-angle relationship

In a triangle, the longest side is opposite the largest angle.

Step2: Find the largest angle

Given angles: $\angle F = 55^\circ$, $\angle G = 63^\circ$, $\angle E = 62^\circ$.
Compare angles: $63^\circ > 62^\circ > 55^\circ$, so $\angle G$ is the largest.

Step3: Identify the side opposite $\angle G$

In $\triangle FGE$, side opposite $\angle G$ is $EF$. Wait, no—wait, vertex labels: $\angle G$ is at $G$, so side opposite $\angle G$ is $EF$? Wait, no: vertices are $F$, $G$, $E$. So side opposite $\angle G$ (at $G$) is $EF$? Wait, no: side $EF$ is between $E$ and $F$, side $FG$ is between $F$ and $G$, side $GE$ is between $G$ and $E$. Wait, angle at $G$: between $FG$ and $GE$, so opposite side is $EF$. Angle at $E$: between $GE$ and $EF$, opposite side is $FG$. Angle at $F$: between $EF$ and $FG$, opposite side is $GE$. Wait, I made a mistake. Let's re-express:

• Side opposite $\angle F$ (55°) is $GE$.
• Side opposite $\angle G$ (63°) is $EF$.
• Side opposite $\angle E$ (62°) is $FG$.

Now, largest angle is $\angle G$ (63°), so side opposite is $EF$? Wait, no—wait, 63° is larger than 62° and 55°, so angle at $G$ is 63°, so side opposite is $EF$. Wait, but wait, the options: GE, LMNOP (invalid), EF, FG. Wait, but wait, maybe I mixed up. Wait, let's check again.

Wait, in a triangle, longer side opposite larger angle. So:

• Angle at F: 55°, opposite side GE.
• Angle at G: 63°, opposite side EF.
• Angle at E: 62°, opposite side FG.

So largest angle is 63° (at G), so opposite side is EF. But wait, the options: GE, LMNOP, EF, FG. Wait, but maybe I made a mistake. Wait, no—wait, the angle at F is 55°, angle at G is 63°, angle at E is 62°. So the largest angle is 63° (at G), so the side opposite to 63° is EF. Wait, but let's check the options. The options are GE (red), LMNOP (blue), EF (orange), FG (green). Wait, but maybe I messed up the side labels. Wait, vertex F is top, G is left, E is right. So side GE is the base (between G and E), side FG is left side (F to G), side EF is right side (E to F). So angle at G: between FG and GE, so opposite side is EF. Angle at E: between GE and EF, opposite side is FG. Angle at F: between EF and FG, opposite side is GE. So:

• Angle F (55°) → opposite GE (base)
• Angle G (63°) → opposite EF (right side)
• Angle E (62°) → opposite FG (left side)

So largest angle is 63° (G), so opposite side is EF. Wait, but the red option is GE, orange is EF, green is FG. Wait, but maybe I made a mistake. Wait, no—wait, 63° is larger than 62° and 55°, so the side opposite 63° is EF. So EF should be the longest. Wait, but let's confirm: in a triangle, the longest side is opposite the largest angle. So largest angle is 63° (at G), so side opposite is EF. So the correct option is EF (orange button, labeled "EF"). Wait, but the red button is GE, which is opposite 55°, so GE is opposite the smallest angle, so it's the shortest. Wait, no—wait, 55° is the smallest angle, so opposite side GE is the shortest. Then 62° (angle E) opposite FG, 63° (angle G) opposite EF. So EF is opposite 63°, FG opposite 62°, GE opposite 55°. So EF is the longest. So the answer is EF (the orange button with "EF").

Answer:

The longest side is the one opposite the largest angle (63° at \( G \)). The side opposite \( \angle G \) is \( EF \). Among the options, the correct one is the orange button labeled "EF". So the answer is:

• The option with "EF" (the orange button, labeled "EF").

(Note: The blue option "LMNOP" is invalid, so we ignore it. The red is GE (shortest), green is FG (shorter than EF), orange is EF (longest).)