From the diagram, we see that ∠1 and ∠4 are vertical angles. According to the Vertical Angles Congruence Theorem, vertical angles are congruent. Therefore, since m∠1=105^(∘), we know that m∠4=105^(∘).
To find the measure of ∠5, we can use the fact that ∠5 and ∠4 are alternate interior angles. Since the lines that are cut by the transversal are parallel, we can, by the Alternate Interior Angles Theorem, say that
∠5≅ ∠4.
Similarly, we see that ∠4 and ∠8 are corresponding angles. By the Corresponding Angles Congruence Theorem we know they are congruent: ∠8≅ ∠4. Because m∠4=105^(∘) we must have that
m∠5=105^(∘) and m∠8=105^(∘).
Let's mark these angles in our diagram as well.
