Question

use the given information to prove that $\triangle rst\cong\triangle uvt$.
given: $\angle rst$ is a right angle
$\angle uvt$ is a right angle
$\overline{st}\cong\overline{vt}$
prove: $\triangle rst\cong\triangle uvt$

3 $\overline{st}\cong\overline{vt}$ given
4 $\angle rts\cong\angle utv$ vertical angles property
5 0 reason?

Explanation:

Step1: Identify the right angles

Since \(\angle RST\) is a right angle and \(\angle UVT\) is a right angle, \(\angle RST\cong\angle UVT = 90^{\circ}\) (by the definition of right angles).

Step2: Use the given side - angle - side (SAS) or angle - side - angle (ASA) or other congruence criteria

We know that \(\overline{ST}\cong\overline{VT}\) (given) and \(\angle RTS\cong\angle UTV\) (vertical angles property). Now, using the Angle - Side - Angle (ASA) congruence criterion:
In \(\triangle RST\) and \(\triangle UVT\), \(\angle RST\cong\angle UVT\), \(\overline{ST}\cong\overline{VT}\), \(\angle RTS\cong\angle UTV\).

Answer:

\(\triangle RST\cong\triangle UVT\) by the ASA (Angle - Side - Angle) congruence criterion.