Question

name the marked angle in 2 different ways.

Explanation:

Step1: Identify the vertex and sides

The marked angle has vertex \( B \), and its sides are \( BE \) and \( BC \) (or considering the other side from \( B \) to \( D \), but the marked angle is between \( BE \) and \( BC \), or also can be named using the vertex and the two rays. Another way is using the vertex alone if there's only one angle at \( B \) (but here we have two angles at \( B \), so better to use three points or the vertex with the specific sides). Wait, the marked angle is at \( B \), between \( BE \) and \( BC \), or also between \( B \)'s two segments: one to \( E \) and one to \( C \), or also, the angle can be named as \( \angle EBC \) (using three points: \( E \), \( B \), \( C \)) and \( \angle B \) (but wait, is there only one angle at \( B \)? Wait, no, at \( B \) there are two angles: one between \( BD \) and \( BE \), and one between \( BE \) and \( BC \). Wait, the marked angle is the one between \( BE \) and \( BC \), so first way: \( \angle EBC \) (three - point notation, with \( B \) as the vertex, \( E \) and \( C \) as the points on the sides). Second way: since the vertex is \( B \), and if we consider the angle at \( B \) between \( BE \) and \( BC \), we can also name it as \( \angle B \) (but wait, sometimes when there are multiple angles at a vertex, we use three - point notation, but maybe in this case, the marked angle is \( \angle EBC \) and \( \angle CBE \) (but no, \( \angle EBC \) and \( \angle CBE \) are the same, just reversed. Wait, maybe the other side is \( B \) to \( D \)? No, the marked angle is at \( B \), between \( BE \) and \( BC \). Wait, another way: the angle can be named as \( \angle EBC \) (using the three points in order: \( E \) - \( B \) - \( C \)) and also as \( \angle B \) (if there's no confusion, but actually, at \( B \) there are two angles: one between \( BD \) and \( BE \), and one between \( BE \) and \( BC \). Wait, maybe the marked angle is \( \angle EBC \) and \( \angle CBE \) (but that's the same). Wait, no, maybe the vertex is \( B \), and the two sides are \( BE \) and \( BC \), so first name: \( \angle EBC \) (three - point angle notation, with \( B \) as the vertex, \( E \) and \( C \) on the sides). Second name: since the angle is at vertex \( B \), and if we consider the angle between \( BE \) and \( BC \), we can also name it as \( \angle B \) (but sometimes, when there are multiple angles at a vertex, we use three - point, but maybe in this case, the two ways are \( \angle EBC \) and \( \angle CBE \) (but that's the same). Wait, no, maybe I made a mistake. Wait, the angle is at \( B \), with one side going to \( E \) and one to \( C \), so the angle can be named as \( \angle EBC \) (using the middle letter as the vertex) and also as \( \angle B \) (if there's only one angle at \( B \), but here there are two angles at \( B \): one between \( BD \) and \( BE \), and one between \( BE \) and \( BC \). Wait, maybe the marked angle is \( \angle EBC \) and \( \angle CBE \) (but that's the same angle, just the order of the points is reversed). Alternatively, maybe the other side is \( B \) to \( D \)? No, the marked angle is the one with the arc, which is between \( BE \) and \( BC \). So the two ways are:

1. Using three points: \( \angle EBC \) (vertex \( B \), with \( E \) and \( C \) on the sides)
2. Using the vertex: \( \angle B \) (but only if there's no ambiguity, but in this case, since there are two angles at \( B \), maybe the three - point and the angle with the vertex and the two sides. Wait, another way: the angle can be named as \( \angle EBC \) and \( \angle CBE \) (but that's the same). Wait, no, perhaps the correct two ways are \( \angle EBC \) (three - point) and \( \angle B \) (vertex - only), assuming that in the context, the marked angle is the one between \( BE \) and \( BC \), and the other angle at \( B \) (between \( BD \) and \( BE \)) is unmarked. So we can name it as \( \angle EBC \) and \( \angle B \).

Step2: Confirm the angle naming rules

Angle naming rules:
- Three - point notation: \( \angle ABC \) where \( B \) is the vertex, \( A \) and \( C \) are on the sides.
- Vertex - only notation: \( \angle B \) when there's only one angle at vertex \( B \) (or when the context makes it clear which angle is referred to).

In this case, the marked angle has vertex \( B \), with sides \( BE \) and \( BC \). So:

First way: \( \angle EBC \) (three - point, \( E - B - C \))

Second way: \( \angle B \) (vertex - only, since the marked angle is the one between \( BE \) and \( BC \), and if we assume the other angle at \( B \) (between \( BD \) and \( BE \)) is not the marked one, so we can use \( \angle B \) to refer to the marked angle.

Answer:

\( \angle EBC \) and \( \angle B \) (or \( \angle CBE \) and \( \angle EBC \), but \( \angle EBC \) and \( \angle B \) are more appropriate)