Question

1. matching.

order the angles of \\( \triangle def \\) from smallest to largest.
triangle def with sides: de = 5 cm, ef = 3.5 cm, df = 7 cm
dropdown menus and options: a. \\( \angle e \\), b. \\( \angle f \\), c. \\( \angle d \\)

1. matching.

1.
35 mm
2.

Explanation:

Step1: Recall Triangle Angle-Side Relationship

In a triangle, the larger the side length, the larger the angle opposite to it. So we first identify the sides and their opposite angles.
- Side opposite \( \angle D \): \( EF = 3.5 \, \text{cm} \)
- Side opposite \( \angle F \): \( DE = 5 \, \text{cm} \)
- Side opposite \( \angle E \): \( DF = 7 \, \text{cm} \)

Step2: Order the Sides by Length

Order the side lengths from smallest to largest: \( 3.5 \, \text{cm} < 5 \, \text{cm} < 7 \, \text{cm} \) (i.e., \( EF < DE < DF \)).

Step3: Order the Angles by Opposite Sides

Since the angle opposite a shorter side is smaller, we map the sides to their opposite angles:
- Smallest side \( EF = 3.5 \, \text{cm} \) (opposite \( \angle D \)) → smallest angle \( \angle D \)? Wait, no: Wait, side opposite \( \angle D \) is \( EF \) (length 3.5), side opposite \( \angle F \) is \( DE \) (length 5), side opposite \( \angle E \) is \( DF \) (length 7). So:
- \( EF = 3.5 \) (opposite \( \angle D \)) → \( \angle D \) is opposite the smallest side, so \( \angle D \) is the smallest? Wait no, wait: Wait, side \( EF \) is opposite \( \angle D \), side \( DE \) is opposite \( \angle F \), side \( DF \) is opposite \( \angle E \). So:
- \( EF = 3.5 \, \text{cm} \) (opposite \( \angle D \))
- \( DE = 5 \, \text{cm} \) (opposite \( \angle F \))
- \( DF = 7 \, \text{cm} \) (opposite \( \angle E \))

So ordering the sides: \( EF (3.5) < DE (5) < DF (7) \). Therefore, the angles opposite them will be in the same order: \( \angle D \) (opposite \( EF \)) < \( \angle F \) (opposite \( DE \)) < \( \angle E \) (opposite \( DF \))? Wait, no: Wait, angle opposite the smallest side is the smallest angle. So smallest side is \( EF = 3.5 \), opposite \( \angle D \), so \( \angle D \) is smallest? Wait, no, wait: Let's label the triangle correctly. In \( \triangle DEF \), vertices are D, E, F. So:
- \( DE = 5 \, \text{cm} \) (between D and E)
- \( EF = 3.5 \, \text{cm} \) (between E and F)
- \( DF = 7 \, \text{cm} \) (between D and F)

So:
- Angle at D (\( \angle D \)) is between \( DE \) and \( DF \), so its opposite side is \( EF \) (length 3.5)
- Angle at F (\( \angle F \)) is between \( EF \) and \( DF \), so its opposite side is \( DE \) (length 5)
- Angle at E (\( \angle E \)) is between \( DE \) and \( EF \), so its opposite side is \( DF \) (length 7)

So side lengths: \( EF = 3.5 \), \( DE = 5 \), \( DF = 7 \). So from smallest to largest: \( EF < DE < DF \). Therefore, the angles opposite them: \( \angle D \) (opposite \( EF \)) < \( \angle F \) (opposite \( DE \)) < \( \angle E \) (opposite \( DF \)). Wait, that would mean \( \angle D < \angle F < \angle E \). Wait, let's check again. Wait, side opposite \( \angle D \): in triangle, side opposite angle D is EF (since angle D is at vertex D, so the side not connected to D is EF). Yes. Side opposite angle F is DE (side not connected to F is DE). Side opposite angle E is DF (side not connected to E is DF). So:

- \( EF = 3.5 \) (opposite \( \angle D \))
- \( DE = 5 \) (opposite \( \angle F \))
- \( DF = 7 \) (opposite \( \angle E \))

So the lengths of the sides opposite the angles: \( 3.5 < 5 < 7 \). Therefore, the angles themselves (since larger side → larger angle) will be ordered as: \( \angle D < \angle F < \angle E \), because the side opposite \( \angle D \) is the smallest, so \( \angle D \) is the smallest; then side opposite \( \angle F \) is next, so \( \angle F \) is next; then side opposite \( \angle E \) is largest, so \( \angle E \) is largest.

Wait, but let's confirm with the side lengths: \( DE = 5 \), \( EF = 3.5 \), \( DF = 7 \). So the sides adjacent to angle D: DE (5) and DF (7), so opposite side is EF (3.5). Sides adjacent to angle F: EF (3.5) and DF (7), opposite side is DE (5). Sides adjacent to angle E: DE (5) and EF (3.5), opposite side is DF (7). So yes, opposite sides: \( \angle D \) ↔ \( EF = 3.5 \), \( \angle F \) ↔ \( DE = 5 \), \( \angle E \) ↔ \( DF = 7 \). So ordering the sides: \( 3.5 < 5 < 7 \), so angles: \( \angle D < \angle F < \angle E \). Wait, but let's check the side lengths again. Wait, the problem's triangle: DE is 5 cm, EF is 3.5 cm, DF is 7 cm. So the sides are: DE = 5, EF = 3.5, DF = 7. So the order of sides from smallest to largest: EF (3.5) < DE (5) < DF (7). Therefore, the angles opposite them: angle D (opposite EF) < angle F (opposite DE) < angle E (opposite DF). So the angles from smallest to largest are \( \angle D \), \( \angle F \), \( \angle E \). Wait, but let's check the options: the options are a. \( \angle E \), b. \( \angle F \), c. \( \angle D \). So the order from smallest to largest: c (∠D), b (∠F), a (∠E).

Answer:

Smallest: c. \( \angle D \), Middle: b. \( \angle F \), Largest: a. \( \angle E \) (So the order from smallest to largest is \( \angle D \), \( \angle F \), \( \angle E \))