Question

use the given information to prove that ( \triangle rst cong \triangle vut ).
given: ( overline{rt} cong overline{vt} )
( angle rst cong angle vut )
prove: ( \triangle rst cong \triangle vut )
1 ( overline{rt} cong overline{vt} ) given
2 ( angle rst cong angle vut ) given
3 ( \triangle rst cong \triangle vut ) reason?

Explanation:

Step1: Identify the given information

We are given that \(\overline{RT}\cong\overline{VT}\) (a pair of sides) and \(\angle RST\cong\angle VUT\) (a pair of angles). Also, \(\angle RTS\) and \(\angle VTU\) are vertical angles. By the vertical - angle theorem, \(\angle RTS\cong\angle VTU\).

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

The AAS (Angle - Angle - Side) 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 \(\triangle RST\) and \(\triangle VUT\):

• \(\angle RST\cong\angle VUT\) (given)
• \(\angle RTS\cong\angle VTU\) (vertical angles)
• \(\overline{RT}\cong\overline{VT}\) (given)

Answer:

The reason for \(\triangle RST\cong\triangle VUT\) is the AAS (Angle - Angle - Side) congruence criterion.