EXPLANATION :
Cases in which NO triangle can be constructed :
1. The sum of interir angles ins NOT equal to 180 degrees.
2. If the sides of the triangle DOES NOT conform with :
"Any side of the triangle must be greater than the difference of the other two sides."
and
"Any side of the triangle must be less than the sum of the other two sides."
Cases in which exactly ONE triangle can be constructed :
1. Three sides of the triangle are given.
2. 2 angles and an included side are given.
3. 2 sides and an included angle are given.
Cases in which MANY triangles can be constructed :
1. Three of the angles are given.
Now let's proceed with the given problem.
Condition A :
The three sides of te triangle are given ()4 inches, 8 inches and 10 inches
So it falls under ONE Triangle.
Condition B :
The interior angles are given, 60, 40 and 90.
The sum of the interior angles is 60 + 40 + 90 = 190
Which is NOT equal to 180 egrees.
So it falls under NO Triangle.
Condition C :
2 sides of the triangle and an included angle is given.
So it falls under ONE Triangle
Condition D :
Three sides are given, 5 inches, 7 inches and 12 inches.
We need to check the sides if it conform with the condition of NO Triangle.
"Any of the sides is greater than the difference of the other two sides."
Let's say 5, and the difference of the other two is 12 - 7 = 5
5 is NOT greater than 5.
So it DOES NOT conform with the property of the triangle.
So it falls under NO Triange.
Cndition E :
Two angles and an included side are given.
So it falls under ONE Triangle
Condition F :
Three angles are given, we need to check the sum of them.
40 + 80 + 60 = 180
Since it is 180 degrees, it can construct a triangle.
If three angles are given, it falls under MANY triangles