Question

complete the proof that \\( \triangle uvw \cong \triangle utx \\).

statement	reason
2 \\( \angle tux \cong \angle vuw \\)	given
3 \\( \overline{tx} \perp \overline{tv} \\)	given
4 \\( \overline{vw} \cong \overline{tx} \\)	asa
5 \\( \angle t \cong \angle v \\)	definition of congruence
6 \\( \triangle uvw \cong \triangle utx \\)	aas additive property of angle measure additive property of length all right angles are congruent alternate interior angles theorem angles forming a linear pair sum to \\( 180^{\circ} \\) asa corresponding angles theorem cpctc definition of angle bisector definition of congruence definition of equilateral triangle definition of midpoint given properties of addition, subtraction, multiplication, and division reflexive property of congruence

Explanation:

Step1: Analyze the given information

We know that \( \overline{VW}\perp\overline{TV}\) (so \( \angle V = 90^{\circ}\)), \( \angle TUX\cong\angle VUW\), \( \overline{TX}\perp\overline{TV}\) (so \( \angle T=90^{\circ}\)), and \( \overline{VW}\cong\overline{TX}\).

Step2: Recall the AAS (Angle - Angle - Side) congruence criterion

The AAS congruence criterion states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.
In \( \triangle UVW\) and \( \triangle UTX\):

• We have \( \angle V\cong\angle T\) (since \( \overline{VW}\perp\overline{TV}\) and \( \overline{TX}\perp\overline{TV}\), all right angles are congruent, so \( \angle V = \angle T=90^{\circ}\))
• \( \angle VUW\cong\angle TUX\) (given)
• \( \overline{VW}\cong\overline{TX}\) (given)

Answer:

AAS