Exercise 25 Page 459

Let's begin by highlighting the given information in the given diagram.

Since XU ∥ VW and UW is a transversal, by the Alternate Interior Angles Theorem we obtain that ∠ XUZ ≅ ∠ VWZ. Besides this, we can see that ∠ XZU and ∠ VZW are vertical angles, and so ∠ XZU ≅ ∠ VZW.

By the Angle-Angle-Side (AAS) Congruence Postulate we get that △ XUZ ≅ △ VWZ, which implies that XZ ≅ VZ. Also, by the Reflexive Property of Congruent Segments we have WZ≅ WZ.

Finally, by applying the Vertical Angles Theorem again we obtain that m ∠ VZW = m ∠ XZU and m ∠ XZW = m ∠ UZV. Substituting them into the inequality above, we will obtain the required result. cc m ∠ VZW > m ∠ XZW & ⇓ & m ∠ XZU > m ∠ UZV & ✓

Two-Column Proof

We will summarize the proof we did before in the following two-column table.