a From the diagram we see that the triangles have two pairs of congruent angles. We also see that AC is a shared side, which means this side is congruent in our two triangles.
Also, notice that ∠ QSP and ∠ QSR form a straight angle pair, which means they sum to 180^(∘). Since one angle is 90^(∘), the other one must be 90^(∘) as well.
With this information, we can claim congruence between △ SQP and △ SQR by the AAS Congruence Condition.
△ SQP ≅ △ SQR
c In addition to the congruent angle pair we see in the diagram, the pair of angles at M form vertical angles, which means they are congruent according to the Vertical Angles Theorem.
Notice though that we have no information about the length of the triangles sides. We need at least one pair of corresponding sides to be of the same length in order to claim congruence between the triangles. Therefore, we do not have enough information to claim congruence.
d If Y is the midpoint of WT and XZ, it means it cuts those segments in two halves of equal lengths. Let's add this information to the diagram.
In addition to the two pairs of congruent sides, the pair of angles at Y form vertical angles, which means they are congruent according to theVertical Angles Theorem.
Additionally, the two triangles share EG as a side, which means this side is congruent in our triangles according to the Reflexive Property of Congruence.
f We have not been given any information about the triangles' angles. Therefore, the only congruence property we could hope to use is the SSS (Side-Side-Side) Congruence Condition. From the diagram, we notice that the triangles have two pairs of sides with equal length.
As for the third side, since AF=CD and both triangles share FC, we know that the third side must be congruent as well. If we label the length of FC as a, we can show that the triangles have three pairs of congruent sides.
With this information, we can claim congruence between △ ABC and △ DEF by the SSS Congruence Condition.
△ ABC ≅ △ DEF
