To determine which angles can be trisected using a straightedge and compass, we need to understand the principles of classical Greek geometry, specifically which angles meet the constructible angle criteria.
The process of trisecting an angle involves dividing an angle into three equal smaller angles. Not all angles are trisectable using only a straightedge and compass.
We are given four sets of angles to choose from:
A. [tex]\( 90^\circ, 30^\circ \)[/tex]
B. [tex]\( 75^\circ, 90^\circ \)[/tex]
C. [tex]\( 90^\circ, 45^\circ \)[/tex]
D. [tex]\( 60^\circ, 45^\circ \)[/tex]
Let's analyze these options:
1. Option A: [tex]\( 90^\circ \)[/tex]; [tex]\( 30^\circ \)[/tex]
- [tex]\( 90^\circ \)[/tex] is trisectable because it can be divided into three [tex]\( 30^\circ \)[/tex] angles.
- [tex]\( 30^\circ \)[/tex] is already a simple angle and thus cannot be trisected to yield integer degrees.
2. Option B: [tex]\( 75^\circ \)[/tex]; [tex]\( 90^\circ \)[/tex]
- [tex]\( 75^\circ \)[/tex] is not trisectable using classical straightedge and compass methods.
- [tex]\( 90^\circ \)[/tex] is trisectable (as previously noted).
3. Option C: [tex]\( 90^\circ \)[/tex]; [tex]\( 45^\circ \)[/tex]
- [tex]\( 90^\circ \)[/tex] is trisectable.
- [tex]\( 45^\circ \)[/tex] is not trisectable into smaller integer degrees using straightedge and compass.
4. Option D: [tex]\( 60^\circ \)[/tex]; [tex]\( 45^\circ \)[/tex]
- [tex]\( 60^\circ \)[/tex] is trisectable because it can be divided into three [tex]\( 20^\circ \)[/tex] angles.
- [tex]\( 45^\circ \)[/tex] is not trisectable into smaller integer degrees using straightedge and compass.
From this analysis, the angles that are confirmed to be trisectable with a straightedge and compass are [tex]\( 60^\circ \)[/tex] and [tex]\( 90^\circ \)[/tex], which points to Option D.
So, the two angles that can be trisected with a straightedge and compass are [tex]\( 60^\circ \)[/tex] and [tex]\( 90^\circ \)[/tex].
The correct answer is:
[tex]\[ \boxed{60^\circ \text{ and } 90^\circ} \][/tex]
