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:

Step1: Recall angle - naming rule

An angle is named using three points with the vertex in the middle. For angles with vertex \(V\), we list all possible combinations of rays with \(V\) as the starting - point.

Step2: Analyze each diagram

For example, in the first diagram with rays \(VH\), \(VI\), and \(VJ\), the angles are \(\angle HVJ\), \(\angle HV I\), \(\angle IVJ\). We do this for all diagrams.

1. \(\angle HVJ\), \(\angle HV I\), \(\angle IVJ\)
2. \(\angle DV E\), \(\angle DV F\), \(\angle EV F\)
3. \(\angle RV S\), \(\angle RV T\), \(\angle SV T\)
4. \(\angle DV E\), \(\angle DV F\), \(\angle EV F\)
5. \(\angle KVL\), \(\angle KVM\), \(\angle LVM\)
6. \(\angle FVG\), \(\angle FVH\), \(\angle FVI\), \(\angle GVH\), \(\angle GVI\), \(\angle HVI\)
7. \(\angle FVG\), \(\angle FVH\), \(\angle FVI\), \(\angle GVH\), \(\angle GVI\), \(\angle HVI\)
8. \(\angle CV D\), \(\angle CV E\), \(\angle CV F\), \(\angle DV E\), \(\angle DV F\), \(\angle EV F\)

1. Exterior
2. Interior
3. Exterior
4. On the angle
5. Interior
6. Exterior

27)

1. First, draw an acute angle \(\angle ABC\). An acute angle has a measure between \(0^{\circ}\) and \(90^{\circ}\).
2. Then, draw an obtuse angle \(\angle DBE\) such that point \(D\) is inside \(\angle ABC\). Place point \(B\) as the common vertex. Draw ray \(BA\) and \(BC\) to form \(\angle ABC\). Then draw ray \(BD\) and \(BE\) such that \(\angle DBE> 90^{\circ}\) and \(D\) lies within the region of \(\angle ABC\).

1. If \(\angle ABC\) is an acute angle and \(\angle DBE\) is an obtuse angle with a common vertex \(B\), assume \(\angle ABC=\alpha\) (\(0 <\alpha<90^{\circ}\)) and \(\angle DBE = \beta\) (\(90^{\circ}<\beta<180^{\circ}\)). If \(D\) is in the interior of \(\angle ABC\), and we consider the angular regions, the obtuse - angle \(\angle DBE\) will extend beyond the region of the acute - angle \(\angle ABC\). So, if \(D\) is in the interior of \(\angle ABC\), then for point \(E\) (on the non - common ray of \(\angle DBE\) with respect to \(D\)), it cannot be in the interior of \(\angle ABC\) because the obtuse - angle \(\angle DBE\) will cover an area that is outside of the acute - angle \(\angle ABC\) region.

Answer:

1. \(\angle HVJ\), \(\angle HV I\), \(\angle IVJ\)
2. \(\angle DV E\), \(\angle DV F\), \(\angle EV F\)
3. \(\angle RV S\), \(\angle RV T\), \(\angle SV T\)
4. \(\angle DV E\), \(\angle DV F\), \(\angle EV F\)
5. \(\angle KVL\), \(\angle KVM\), \(\angle LVM\)
6. \(\angle FVG\), \(\angle FVH\), \(\angle FVI\), \(\angle GVH\), \(\angle GVI\), \(\angle HVI\)
7. \(\angle FVG\), \(\angle FVH\), \(\angle FVI\), \(\angle GVH\), \(\angle GVI\), \(\angle HVI\)
8. \(\angle CV D\), \(\angle CV E\), \(\angle CV F\), \(\angle DV E\), \(\angle DV F\), \(\angle EV F\)
9. Exterior
10. Interior
11. Exterior
12. On the angle
13. Interior
14. Exterior
15. See above explanation for drawing steps
16. See above explanation for reasoning