Tonya's proof is the correct one, while Pearl's proof is incorrect because it makes an invalid assumption about adjacent angles.
The two proofs shown in the image are trying to prove that alternate interior angles, which are formed when a transversal cuts two parallel lines, are congruent. Both proofs reach the desired conclusion, but they take different approaches.
Tonya's proof starts by identifying two vertical angles. Vertical angles are known to be congruent, so this step is valid. She then uses the Corresponding Angles Theorem to show that two other pairs of angles are congruent. Finally, she uses the Transitive Property to conclude that the desired alternate interior angles are congruent.
Pearl's proof starts by identifying two adjacent angles. Adjacent angles are not necessarily congruent, so this step is not valid. However, she then uses the Corresponding Angles Theorem to show that two other pairs of angles are congruent. Finally, she uses the Transitive Property to conclude that the desired alternate interior angles are congruent.
Therefore, Pearl's proof is the one that is incorrect. Her mistake is assuming that adjacent angles are congruent, which is not always the case. Tonya's proof, on the other hand, is correct and does not rely on any false assumptions.