To determine which angles can be trisected using a straightedge and compass, we need to understand the concept of angle trisection. Trisecting an angle means dividing it into three equal parts. This is a classic problem in geometry with certain constraints: not all angles can be trisected using classical geometric constructions.
Considering the choices given in the problem, let’s analyze each pair:
Choice A: [tex]$90^{\circ} ; 45^{\circ}$[/tex]
- A [tex]$90^{\circ}$[/tex] angle can be trisected into three [tex]$30^{\circ}$[/tex] angles.
- A [tex]$45^{\circ}$[/tex] angle can be trisected into three [tex]$15^{\circ}$[/tex] angles.
This pair matches the criteria for angles that can be trisected.
Choice B: [tex]$60^{\circ} ; 45^{\circ}$[/tex]
- A [tex]$60^{\circ}$[/tex] angle cannot be exactly trisected using just a straightedge and compass.
- A [tex]$45^{\circ}$[/tex] angle can be trisected into three [tex]$15^{\circ}$[/tex] angles.
Therefore, this choice does not fully meet the criteria.
Choice C: [tex]$75^{\circ} ; 90^{\circ}$[/tex]
- A [tex]$75^{\circ}$[/tex] angle cannot be exactly trisected using just a straightedge and compass.
- A [tex]$90^{\circ}$[/tex] angle can be trisected into three [tex]$30^{\circ}$[/tex] angles.
Therefore, this choice also does not fully meet the criteria.
Choice D: [tex]$90^{\circ} ; 30^{\circ}$[/tex]
- A [tex]$90^{\circ}$[/tex] angle can be trisected into three [tex]$30^{\circ}$[/tex] angles.
- A [tex]$30^{\circ}$[/tex] angle would trisect into three [tex]$10^{\circ}$[/tex] angles.
This pair doesn’t match because the [tex]$30^{\circ}$[/tex] angle doesn't trisect in a practical geometric construction setting.
Based on these analyses, the correct answer is:
A. [tex]$90^{\circ} ; 45^{\circ}$[/tex]
