Exercise 25 Page 459

Prove that △ XUZ ≅ △ VWZ, and then apply the Converse of the Hinge Theorem to △ XZW and △ VZW. Finally, apply the Vertical Angles Theorem.

Statements	Reasons
1. XU≅ VW, VW > XW, XU∥ VW	1. Given
2. ∠ XUZ ≅ ∠ VWZ	2. Alternate Interior Angles Theorem
3. ∠ XZU ≅ ∠ VZW	3. Vertical Angles Theorem
4. △ XUZ ≅ △ VWZ	4. AAS Congruence Postulate
5. XZ ≅ VZ	5. Definition of congruent polygons
6. WZ ≅ WZ	6. Reflexive Property of Congruent Segments
7. m∠ VZW > m∠ XZW	7. Converse of the Hinge Theorem
8. ∠ VZW ≅ ∠ XZU and ∠ XZW ≅ ∠ UZV	8. Vertical Angles Theorem
9. m∠ VZW = m∠ XZU and m∠ XZW = m∠ UZV	9. Definition of congruent angles
10. m∠ XZU > m∠ UZV	10. Substitution

Practice makes perfect

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.

Statements	Reasons
1. XU≅ VW, VW > XW, XU∥ VW	1. Given
2. ∠ XUZ ≅ ∠ VWZ	2. Alternate Interior Angles Theorem
3. ∠ XZU ≅ ∠ VZW	3. Vertical Angles Theorem
4. △ XUZ ≅ △ VWZ	4. AAS Congruence Postulate
5. XZ ≅ VZ	5. Definition of congruent polygons
6. WZ ≅ WZ	6. Reflexive Property of Congruent Segments
7. m∠ VZW > m∠ XZW	7. Converse of the Hinge Theorem
8. ∠ VZW ≅ ∠ XZU and ∠ XZW ≅ ∠ UZV	8. Vertical Angles Theorem
9. m∠ VZW = m∠ XZU and m∠ XZW = m∠ UZV	9. Definition of congruent angles
10. m∠ XZU > m∠ UZV	10. Substitution

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Two-Column Proof