Question

triangle mno and triangle tuo are drawn below. if \\(\overline{mn} \cong \overline{tu}\\), \\(\angle m \cong \angle t\\), and \\(\angle mon \cong \angle tou\\), use the dropdown box below to determine a transformation that maps triangle mno onto triangle tuo. then use a congruence statement to explain why this is possible.

a rotation counterclockwise about point o until \\(\overline{mn}\\) has the same slope as \\(\overline{tu}\\).

answer attempt 2 out of 2

triangle mno maps onto triangle tuo. this is possible because \\(\triangle mno \cong \triangle tuo\\) by dropdown with options sas, asa, sss, aas, and one congruent figure can be mapped onto another using dropdown with option rigid motions.

Explanation:

Step1: Recall congruence criteria

• ASA (Angle - Side - Angle): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
• We are given that \(\overline{MN}\cong\overline{TU}\) (a side), \(\angle M\cong\angle T\) (an angle), and \(\angle MON\cong\angle TOU\). Since \(\angle MON\) and \(\angle NOU\) form a linear pair, and \(\angle TOU\) and \(\angle UON\) form a linear pair, and \(\angle MON\cong\angle TOU\), we can find that \(\angle MNO\cong\angle TUO\) (because the sum of angles in a triangle is \(180^{\circ}\), and \(\angle MON + \angle M+\angle MNO=180^{\circ}\), \(\angle TOU+\angle T+\angle TUO = 180^{\circ}\)). The side \(\overline{MN}\) is included between \(\angle M\) and \(\angle MNO\), and the side \(\overline{TU}\) is included between \(\angle T\) and \(\angle TUO\).

Answer:

\(\triangle MNO\cong\triangle TUO\) by \(ASA\) (Angle - Side - Angle) congruence criterion.