To determine which set of angles could not form a valid triangle, we need to check if the sum of the angles in each set equals [tex]\(180^\circ\)[/tex]. The sum of the interior angles of a triangle must always be [tex]\(180^\circ\)[/tex].
Let's evaluate each set:
### Set A
Angles:
[tex]\[ m \angle D = 90^\circ, \, m \angle E = 45^\circ, \, m \angle F = 45^\circ \][/tex]
Calculation:
[tex]\[ 90^\circ + 45^\circ + 45^\circ = 180^\circ \][/tex]
Since the sum is [tex]\(180^\circ\)[/tex], these angles could form a valid triangle.
### Set B
Angles:
[tex]\[ m \angle D = 90^\circ, \, m \angle E = 46^\circ, \, m \angle F = 46^\circ \][/tex]
Calculation:
[tex]\[ 90^\circ + 46^\circ + 46^\circ = 182^\circ \][/tex]
Since the sum is [tex]\(182^\circ\)[/tex], which is not [tex]\(180^\circ\)[/tex], these angles could not form a valid triangle.
### Set C
Angles:
[tex]\[ m \angle D = 100^\circ, \, m \angle E = 50^\circ, \, m \angle F = 30^\circ \][/tex]
Calculation:
[tex]\[ 100^\circ + 50^\circ + 30^\circ = 180^\circ \][/tex]
Since the sum is [tex]\(180^\circ\)[/tex], these angles could form a valid triangle.
### Set D
Angles:
[tex]\[ m \angle D = 91^\circ, \, m \angle E = 47^\circ, \, m \angle F = 42^\circ \][/tex]
Calculation:
[tex]\[ 91^\circ + 47^\circ + 42^\circ = 180^\circ \][/tex]
Since the sum is [tex]\(180^\circ\)[/tex], these angles could form a valid triangle.
### Conclusion
The set of angles that could not form a valid triangle is:
[tex]\[ \boxed{B} \][/tex]
