Question

which is the last step of the proof?
uv ⊥ st and us ≅ ut
∠uvs and ∠uvt are right angles
△uvs and △uvt are right triangles
uv ≅ uv
?
?
statement: △uvs ≅ △uvt reason: hl
statement: △uvs ≅ △uvt reason: sas
statement: △uvs ≅ △uvt reason: asa
statement: △uvs ≅ △uvt reason: sss

Explanation:

Step1: Analyze the given information

We have two right - triangles \(\triangle UVS\) and \(\triangle UVT\). We know that \(US\cong UT\) (given), \(UV\perp ST\) (so \(\angle UVS=\angle UVT = 90^{\circ}\)), and \(UV\cong UV\) (reflexive property).

Step2: Recall the congruence criteria

• HL (Hypotenuse - Leg): For right - triangles, if the hypotenuse and one leg of a right - triangle are congruent to the hypotenuse and one leg of another right - triangle, then the two right - triangles are congruent. Here, \(US\) and \(UT\) are the hypotenuses (\(US\cong UT\)), and \(UV\) is the common leg (\(UV\cong UV\)).
• SAS (Side - Angle - Side): Requires two sides and the included angle. The right angle is not between \(US\) and \(UV\) in a way that would fit SAS for these two triangles.
• ASA (Angle - Side - Angle): Requires two angles and the included side. We only have one known angle (\(90^{\circ}\)) for each triangle.
• SSS (Side - Side - Side): Requires three pairs of congruent sides. We only know two pairs of congruent sides (\(US\cong UT\) and \(UV\cong UV\)).

Answer:

Statement: \(\triangle UVS\cong\triangle UVT\) Reason: HL