Question

name all the angles that have ( v ) as a vertex.
13)
14)
15)
16)
17)
18)
19)
20)
state if the given point is interior, exterior, or on the angle.
21)
22)
23)
24)
25)
26)
critical thinking questions:

1. draw a diagram with an acute angle ( abc ) and an obtuse angle ( dbe ) so that point ( d ) is in the interior of angle ( abc ).
2. in question #27, why is it impossible for both point ( d ) and point ( e ) to be in the interior of angle ( abc )?

Explanation:

An acute angle has a measure between \(0^{\circ}\) and \(90^{\circ}\), while an obtuse angle has a measure between \(90^{\circ}\) and \(180^{\circ}\). If \(\angle ABC\) is acute (\(< 90^{\circ}\)) and \(\angle DBE\) is obtuse (\(>90^{\circ}\)), then for \(\angle DBE\) to exist with \(D\) in the interior of \(\angle ABC\), the ray \(BD\) starts from \(B\) (the vertex of \(\angle ABC\)). If both \(D\) and \(E\) were in the interior of \(\angle ABC\), then \(\angle DBE\) would be formed within \(\angle ABC\). But since \(\angle ABC<90^{\circ}\), a non - zero angle \(\angle DBE\) (with \(D\) and \(E\) both in the interior) would have \(\angle DBE<\angle ABC < 90^{\circ}\), which contradicts the fact that \(\angle DBE\) is obtuse (\(>90^{\circ}\)).

Answer:

Because if \(\angle ABC\) is acute (\(<90^{\circ}\)) and \(\angle DBE\) is obtuse (\(>90^{\circ}\)), having both \(D\) and \(E\) inside \(\angle ABC\) would make \(\angle DBE\) (formed within \(\angle ABC\)) less than \(90^{\circ}\), violating the definition of an obtuse angle.