Question

select the correct answer from the drop-down menu.

triangle \\(abc\\) represents a triangular garden with side lengths \\(ab = 10\text{ m}\\), \\(bc = 8\text{ m}\\), and \\(ac = 15\text{ m}\\).

what is the smallest and largest angle of the garden?

the smallest angle is \\(\angle\\)choose an answer. the largest angle is \\(\angle\\)choose an answer.

Explanation:

Identify side lengths and opposite angles

In any triangle, the side lengths are opposite to their corresponding angles:

• Side \(BC = 8\text{ m}\) is opposite to \(\angle A\).
• Side \(AB = 10\text{ m}\) is opposite to \(\angle C\).
• Side \(AC = 15\text{ m}\) is opposite to \(\angle B\).

Order the side lengths

We compare the given side lengths of \(\triangle ABC\):

$$BC < AB < AC$$

$$8\text{ m} < 10\text{ m} < 15\text{ m}$$

Apply the side-angle relationship

A fundamental geometric property states that the smallest angle is opposite the shortest side, and the largest angle is opposite the longest side.

• The shortest side is \(BC = 8\text{ m}\), so the smallest angle is \(\angle A\).
• The longest side is \(AC = 15\text{ m}\), so the largest angle is \(\angle B\).

Answer:

The smallest angle is <blank>\(\angle A\)</blank>. The largest angle is <blank>\(\angle B\)</blank>.