Question

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find the longest side of △def.

Explanation:

Step1: Find angle at E

The sum of angles in a triangle is \(180^\circ\). Given \(\angle D = 38^\circ\) and \(\angle DFE = 180^\circ - 106^\circ = 74^\circ\) (linear pair). So \(\angle E = 180^\circ - 38^\circ - 74^\circ = 68^\circ\).

Step2: Compare angles

Angles: \(\angle D = 38^\circ\), \(\angle E = 68^\circ\), \(\angle F = 74^\circ\) (wait, no, \(\angle DFE\) is \(74^\circ\), but the angle at F in triangle DEF is \(180 - 106 = 74^\circ\)? Wait, no, the angle at F in \(\triangle DEF\) is adjacent to \(106^\circ\), so it's \(180 - 106 = 74^\circ\). Wait, no, original triangle: vertices D, E, F. So angles: \(\angle D = 38^\circ\), \(\angle F\) (at F) is \(180 - 106 = 74^\circ\) (since the angle outside is \(106^\circ\), so internal is supplementary). Then \(\angle E = 180 - 38 - 74 = 68^\circ\). Wait, no, maybe I messed up. Wait, the angle at F: the diagram shows angle at F (between D, F, E) is adjacent to \(106^\circ\), so linear pair: \(\angle DFE = 180 - 106 = 74^\circ\). Then angles: D=38, F=74, E=180-38-74=68. Wait, but the largest angle is at F? No, wait 74, 68, 38. Wait, no, maybe I made a mistake. Wait, the angle at F: is the \(106^\circ\) an external angle? So in triangle DEF, angle at F is \(180 - 106 = 74^\circ\). Then angles: D=38, E=? Wait, no, maybe the angle at F is \(106^\circ\)? Wait, no, the diagram: point F, with a line FC, so angle between EF and FC is \(106^\circ\), so angle between DF and EF (angle at F in triangle DEF) is \(180 - 106 = 74^\circ\). Then angles: D=38, F=74, E=180-38-74=68. So the largest angle is at F (74°)? Wait, no, 74, 68, 38. Wait, but the side opposite the largest angle is the longest. So angle at F is 74°, angle at E is 68°, angle at D is 38°. Wait, no, wait, maybe I miscalculated. Wait, let's recalculate: sum of angles in triangle is 180. So \(\angle D + \angle E + \angle F = 180\). \(\angle D = 38\), \(\angle F\) (at F) is \(180 - 106 = 74\) (since the angle outside is 106, so internal is 74). Then \(\angle E = 180 - 38 - 74 = 68\). So angles: D=38, E=68, F=74. So the largest angle is at F (74°)? Wait, no, 74 is larger than 68 and 38. Wait, but the side opposite angle D is EF, opposite angle E is DF, opposite angle F is DE. Wait, no: in triangle DEF, side opposite angle D (38°) is EF, side opposite angle E (68°) is DF, side opposite angle F (74°) is DE. Wait, but 74 is the largest angle? Wait, no, 74, 68, 38. So largest angle is at F (74°), so side opposite is DE. Wait, but wait, maybe the angle at F is 106°? Wait, maybe I misread the diagram. If the angle at F (between D, F, E) is 106°, then angles: D=38, F=106, E=180-38-106=36. Then largest angle is at F (106°), so side opposite is DE. Ah! That's probably my mistake. I thought the 106° was external, but maybe it's internal. So if angle at F is 106°, then: \(\angle D = 38^\circ\), \(\angle F = 106^\circ\), so \(\angle E = 180 - 38 - 106 = 36^\circ\). Then the largest angle is at F (106°), so the side opposite is DE (side opposite angle F is DE). Wait, that makes more sense. So I misread the diagram: the angle at F (between D, F, E) is 106°, not external. So that's the error. So correcting: angle at F is 106°, angle at D is 38°, so angle at E is 180 - 38 - 106 = 36°. Then the largest angle is at F (106°), so the side opposite to angle F is DE (since in triangle DEF, side opposite angle F is DE, side opposite angle D is EF, side opposite angle E is DF). So the longest side is opposite the largest angle, so DE. Wait, no: side opposite angle D (38°) is EF, side opposite angle E (36°) is DF, side opposite angle F (106°) is DE. So DE is opposite the largest angle (106°), so DE is the longest side.

Step1: Correct angle calculation

In \(\triangle DEF\), angles: \(\angle D = 38^\circ\), \(\angle F = 106^\circ\) (internal angle at F), so \(\angle E = 180^\circ - 38^\circ - 106^\circ = 36^\circ\).

Step2: Identify longest side

The longest side in a triangle is opposite the largest angle. Largest angle is \(\angle F = 106^\circ\), so side opposite \(\angle F\) is \(DE\) (since in \(\triangle DEF\), side opposite \(\angle F\) is \(DE\), side opposite \(\angle D\) is \(EF\), side opposite \(\angle E\) is \(DF\)).

Answer:

The longest side of \(\triangle DEF\) is \(DE\) (or the side opposite the largest angle, which is \(DE\)). Wait, but let's confirm: angle at F is 106°, which is the largest, so side opposite is DE. So the longest side is \(DE\).