Question

proof write a two - column proof or a paragraph proof.

1. given ▶ ( overline { v w } parallel overline { z y } ),

( overline { u v } cong overline { x w } ), ( overline { u z } cong overline { x y } ),
( overline { v w } perp overline { v z } ), ( overline { v w } perp overline { w y } )
prove ▶ ( angle u cong angle x )

Explanation:

Step1: Analyze the given perpendicular lines

Since \(\overline{VW}\perp\overline{VZ}\) and \(\overline{VW}\perp\overline{WY}\), then \(\angle VZW = \angle WYX=90^{\circ}\) (definition of perpendicular lines).

Step2: Use the parallel - line property

Because \(\overline{VW}\parallel\overline{ZY}\), then \(\angle VZW+\angle UVZ = 180^{\circ}\) and \(\angle WYX+\angle XWY=180^{\circ}\) (consecutive interior angles). Since \(\angle VZW = \angle WYX = 90^{\circ}\), we have \(\angle UVZ=\angle XWY = 90^{\circ}\).

Step3: Apply the Hypotenuse - Leg (HL) congruence criterion

We are given that \(\overline{UV}\cong\overline{XW}\) (hypotenuse) and \(\overline{UZ}\cong\overline{XY}\) (leg). In right - triangles \(\triangle UVZ\) and \(\triangle XWY\), by the HL congruence criterion (\(HL\): if the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent), \(\triangle UVZ\cong\triangle XWY\).

Step4: Conclude the angle congruence

Since \(\triangle UVZ\cong\triangle XWY\), then \(\angle U\cong\angle X\) (corresponding parts of congruent triangles are congruent, \(CPCTC\)).

Answer:

\(\angle U\cong\angle X\) is proved as above.