Question

rotate triangle ( tuv ) counterclockwise about point ( t ) until ( overline{tu} ) has the same slope as ( overline{tb} ). then reflect triangle ( tuv ) over ( overline{tu} ).
answer attempt 1 out of 2
triangle ( tuv ) maps on ( etbc ). this is possible because ( \triangle tuvcong\triangle tbc ) by, and one congruent figure can be mapped onto another using motions.

Explanation:

Step1: Recall Congruence Criteria

• SAS (Side - Angle - Side): Two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle.
• ASA (Angle - Side - Angle): Two angles and the included side of one triangle are equal to two angles and the included side of another triangle.
• SSS (Side - Side - Side): All three sides of one triangle are equal to all three sides of another triangle.
• AAS (Angle - Angle - Side): Two angles and a non - included side of one triangle are equal to two angles and the corresponding non - included side of another triangle.

Since we are dealing with triangle congruence and the problem is about mapping one triangle to another (which is a transformation problem related to congruent triangles), and the general congruence criteria apply.

Step2: Analyze the Transformation

• Rotation and reflection are rigid motions. Rigid motions (isometries) preserve the shape and size of a figure. When we rotate a triangle (which is a rigid motion) and then reflect it (another rigid motion), the two triangles \( \triangle TUV\) and \( \triangle TBC\) are congruent.
• If two triangles are congruent, they satisfy one of the congruence criteria. Without loss of generality, assume that we have established the equality of corresponding parts (sides and angles) through the transformation. For example, if we can show that two sides and the included angle (or other combinations as per the criteria) are equal. But since the problem is more about the concept of congruence and rigid motions (rotation and reflection are rigid motions), and the fact that congruent triangles can be mapped onto each other using rigid (isometric) motions.

Answer:

The first blank: \(SAS\) (or \(ASA\) or \(SSS\) or \(AAS\) depending on the actual side - angle relationships in the full - context of the triangles, but since no specific side - angle data is given beyond the transformation, we assume a general congruence criteria application). The second blank: rigid (rotation and reflection are rigid motions).