a Notice that the triangles have three pairs of congruent sides. This means we can claim congruence by the SSS (Side-Side-Side) Congruence Theorem. However, we also see that the angles between two corresponding sides are different.
Since the included angle of two pairs of corresponding sides is different, these triangles cannot be congruent. Therefore we have a contradiction, which means at least one of these triangles cannot exist.
b Examining the diagram, we see that the triangles have two pairs of congruent sides. We also notice a pair of vertical angles. These are congruent according to the Vertical Angles Theorem.
Since two pairs of sides and their included angle are congruent, these triangles should be congruent according to the SAS (Side-Angle-Side) Congruence Theorem. However, if the triangles are congruent, they should also have three pairs of congruent angles. Using the Triangle Angle Sum Theorem, we can find the remaining angle.
40^(∘)+50^(∘)+ m∠ a = 180^(∘) ⇔ m∠ a =90^(∘)
80^(∘)+50^(∘)+ m∠ b = 180^(∘) ⇔ m∠ b =50^(∘)
As we can see, the triangles do not have three pairs of congruent angles. Therefore, they cannot be congruent.
c Examining the diagram, we notice two smaller triangles inside the big triangle. The smaller triangles share the bigger triangle's height as a side, which means we can claim that this side is congruent according to the Reflexive Property of Congruence.
We also see that the height of the big triangle makes a right angle with the opposite side. Since this angle and its adjacent angle form a linear pair, both angles must be right.
Now we can claim congruence between the smaller triangles by the SAS Congruence Theorem. However, notice that one pair of corresponding sides have different lengths, 12 and 10. Since these sides are not congruent, this triangle is impossible.
d Notice that the triangle is an isosceles triangle. According to the Base Angles Theorem, the base angles should be congruent. However they are not, so this is an impossible triangle.
e Again, notice that the bigger triangle is an isosceles triangle. According to the Base Angles Theorem, the base angles should be congruent. However, they are not, so this is an impossible triangle.
f Examining the diagram, we see that two pairs of sides and the included angle in the triangles are congruent.
This means we can claim that the triangles are congruent by the SAS Congruence Theorem. However, one pair of corresponding sides has different lengths, 13 and 14. Since they are not congruent, this triangle is impossible.
