Answer :
To determine which option is not a valid set of angles for a triangle, we need to verify the sum of the angles for each option. In any valid triangle, the sum of the interior angles must be exactly \(180^\circ\).
Let's examine each set of angles step-by-step:
Option A: \(m \angle D = 90^\circ, m \angle E = 46^\circ, m \angle F = 46^\circ\)
[tex]\[ 90^\circ + 46^\circ + 46^\circ = 182^\circ \][/tex]
The sum is \(182^\circ\), which is more than \(180^\circ\). Thus, this set cannot form a valid triangle.
Option B: \(m \angle D = 90^\circ, m \angle E = 45^\circ, m \angle F = 45^\circ\)
[tex]\[ 90^\circ + 45^\circ + 45^\circ = 180^\circ \][/tex]
The sum is \(180^\circ\), so this is a valid set of angles for a triangle.
Option C: \(m \angle D = 100^\circ, m \angle E = 50^\circ, m \angle F = 30^\circ\)
[tex]\[ 100^\circ + 50^\circ + 30^\circ = 180^\circ \][/tex]
The sum is \(180^\circ\), so this is also a valid set of angles for a triangle.
Option D: \(m \angle D = 91^\circ, m \angle E = 47^\circ, m \angle F = 42^\circ\)
[tex]\[ 91^\circ + 47^\circ + 42^\circ = 180^\circ \][/tex]
The sum is \(180^\circ\), so this is yet another valid set of angles for a triangle.
After checking all the options, the invalid set of angles is found in:
Option A: \(m \angle D = 90^\circ, m \angle E = 46^\circ, m \angle F = 46^\circ\) which sums to \(182^\circ\).
Thus, the answer is [tex]\( \boxed{A} \)[/tex].
Let's examine each set of angles step-by-step:
Option A: \(m \angle D = 90^\circ, m \angle E = 46^\circ, m \angle F = 46^\circ\)
[tex]\[ 90^\circ + 46^\circ + 46^\circ = 182^\circ \][/tex]
The sum is \(182^\circ\), which is more than \(180^\circ\). Thus, this set cannot form a valid triangle.
Option B: \(m \angle D = 90^\circ, m \angle E = 45^\circ, m \angle F = 45^\circ\)
[tex]\[ 90^\circ + 45^\circ + 45^\circ = 180^\circ \][/tex]
The sum is \(180^\circ\), so this is a valid set of angles for a triangle.
Option C: \(m \angle D = 100^\circ, m \angle E = 50^\circ, m \angle F = 30^\circ\)
[tex]\[ 100^\circ + 50^\circ + 30^\circ = 180^\circ \][/tex]
The sum is \(180^\circ\), so this is also a valid set of angles for a triangle.
Option D: \(m \angle D = 91^\circ, m \angle E = 47^\circ, m \angle F = 42^\circ\)
[tex]\[ 91^\circ + 47^\circ + 42^\circ = 180^\circ \][/tex]
The sum is \(180^\circ\), so this is yet another valid set of angles for a triangle.
After checking all the options, the invalid set of angles is found in:
Option A: \(m \angle D = 90^\circ, m \angle E = 46^\circ, m \angle F = 46^\circ\) which sums to \(182^\circ\).
Thus, the answer is [tex]\( \boxed{A} \)[/tex].
