Question

complete the statement:
\\( \triangle tns \\) and \\( \triangle uhs \\) are
congruent by aas.
congruent by sas.
congruent by sss.
not congruent triangles.
show your work

Explanation:

Step1: Identify vertical angles

$\angle TSN=\angle USH$ (vertical angles are equal).

Step2: Check AAS condition

In $\triangle TNS$ and $\triangle UHS$, we have $\angle T=\angle U$ (given as angles in the triangles), $\angle N=\angle H = 90^{\circ}$ (right angles), and $\angle TSN=\angle USH$ (from Step 1). Also, the side opposite one of the angles (the non - included side) can be considered. By the Angle - Angle - Side (AAS) congruence criterion, if two angles and a non - included side of one triangle are equal to the corresponding two angles and non - included side of another triangle, the triangles are congruent.

Answer:

congruent by AAS.