Question

O M N T U A rotation counterclockwise about point OO until start overline, M, N, end overline MN has the same slope as start overline, T, U, end overline TU . Answer Attempt 2 out of 2 Triangle M, N, OMNO maps onto triangle T, U, OTUO. This is possible because triangle, M, N, O, \cong, triangle, T, U, O△MNO≅△TUO by ASA , and one congruent figure can be mapped onto another using rigid motions.

Answer 1

Your statement appears to be regarding a geometric transformation involving triangles $\mathrm{△}MNO$ and $\mathrm{△}TUO$, particularly focusing on the relationship between their slopes when subjected to a counterclockwise rotation.

To clarify your thought process, you seem to be claiming that:

1. Counterclockwise Rotation: You are performing a counterclockwise rotation of triangle $MNO$ about point $O$ until the line segment $\overline{MN}$ has the same slope as line segment $\overline{TU}$.

2. Congruence: You have stated that $\mathrm{△}MNO\cong \mathrm{△}TUO$ by the ASA (Angle-Side-Angle) postulate.

3. Rigid Motions: It is true that congruent figures can be mapped onto each other through rigid motions, including rotations, reflections, and translations.

To ensure clarity in your explanation, consider rephrasing the statement to emphasize these key points:

• The transformation involves a counterclockwise rotation of triangle $MNO$ about point $O$ to align the slope of $\overline{MN}$ with that of $\overline{TU}$.

• The congruence $\mathrm{△}MNO\cong \mathrm{△}TUO$ can be established using the ASA criterion. This congruence implies that their corresponding sides and angles are equal.

• Because the triangles are congruent, you can indeed map triangle $MNO$ onto triangle $TUO$ using rigid motions, thus affirming that their respective line segments will align as described.

The relationship between the slopes before and after the rotation indicates that the triangles are geometrically similar in this aspect, reinforcing the original statement of congruence and the validity of the transformation.

If this was your intent, then it seems you've constructed a logical argument based on the properties of congruent triangles and geometric transformations!