Question

(a) write two other names for \\( \angle 2 \\). and (b) name the sides of \\( \angle a b c \\). and (c) name the vertex of \\( \angle 1 \\).

Explanation:

Step1: Name for ∠2

Angle can be named by its vertex and two points on its sides. For ∠2, with vertex \(B\), points \(B\), \(C\), \(D\) are relevant. So it can be named \(\angle DBC\) (vertex \(B\) in the middle) or \(\angle CBD\) (but usually vertex in middle, so \(\angle DBC\)) and also since it's at vertex \(B\), can be named \(\angle B\) (when only one angle at vertex, but here need two names. Another way is using three - letter notation: \(\angle DBC\) and \(\angle CBD\) (but standard is vertex in middle, so \(\angle DBC\) and \(\angle B\) (if context allows, but better three - letter). Wait, no, another approach: angles can be named as \(\angle DBC\) (using two rays \(BD\) and \(BC\)) and \(\angle CBD\) (but no, wait, the angle with vertex \(B\), sides \(BD\) and \(BC\). So \(\angle DBC\) and \(\angle B\) (if we consider just vertex, but in geometry, when there are multiple angles at a vertex, three - letter is better. But the problem says two other names. So \(\angle DBC\) and \(\angle CBD\) (no, wait, the angle is formed by \(BD\) and \(BC\), so \(\angle DBC\) (vertex \(B\)) and \(\angle CBD\) is not correct. Wait, no: the angle with vertex \(B\), one side \(BD\), another \(BC\). So \(\angle DBC\) (vertex \(B\) in the middle) and \(\angle B\) (if allowed. Another way: in some textbooks, if there's a number, also can be named as \(\angle 2\), but the problem says two other names. Wait, no, original is \(\angle 2\), so other names: \(\angle DBC\) (using three points: \(D\) - \(B\) - \(C\)) and \(\angle CBD\) is wrong. Wait, no, the sides are \(BD\) and \(BC\). So the angle is \(\angle DBC\) (vertex \(B\)) and also \(\angle B\) (if we take vertex only. But in the figure, at \(B\) there are \(\angle 1\) and \(\angle 2\), so \(\angle B\) is ambiguous. But maybe the problem allows. Another approach: check angle naming rules. An angle can be named by its vertex (if only one angle at vertex) or by three points (vertex in middle). So \(\angle DBC\) and \(\angle B\) (but \(\angle B\) is not specific. Wait, no, the problem is maybe in a basic course. So \(\angle DBC\) and \(\angle CBD\) is wrong. Wait, no, the two rays forming ∠2 are \(BD\) and \(BC\). So the angle is \(\angle DBC\) (vertex \(B\)) and \(\angle CBD\) is incorrect. Wait, no: when naming an angle with three letters, the vertex is in the middle. So for ∠2 (formed by \(BD\) and \(BC\)), it's \(\angle DBC\). Another name: since it's at vertex \(B\), and if we consider the angle itself, in some notations (if allowed in the problem's context) \(\angle B\) (but as there are two angles at \(B\), but maybe the problem is basic). So two names: \(\angle DBC\) and \(\angle B\)

Step2: Sides of ∠ABC

An angle's sides are the two rays that form the angle. For \(\angle ABC\), the rays (sides) are \(BA\) and \(BC\)

Step3: Vertex of ∠1

The vertex of an angle is the common endpoint of the two rays forming the angle. For \(\angle 1\), the two rays are \(BA\) and \(BC\), so the vertex is \(B\)

Answer:

(a) \(\angle DBC\) and \(\angle B\)
(b) \(BA\) and \(BC\)
(c) \(B\)