Question

2 what is the definition of congruence? a if two figures have the same shape, then they are congruent. b if two figures have the same area, then they are congruent. c if there is a sequence of transformations taking one figure to another, then they are congruent. d if there is a sequence of rotations, reflections, and translations that take one figure to the other, then they are congruent. 3 consider the two triangles in the following diagram. there is a sequence of rigid transformations that takes a to a, b to b, and c to c. the same sequence takes d to d. draw and label d.

Explanation:

Step1: Recall congruence definition

Two figures are congruent if there is a sequence of rotations, reflections, and translations (rigid - transformations) that map one figure onto the other.

Step2: Analyze options

Option A is wrong because having the same shape is not the full - definition of congruence. Option B is wrong as having the same area does not imply congruence. Option C is the correct definition of congruence. Option D is a restatement of the correct concept in a different way but Option C is the standard definition form.

Answer:

C. If there is a sequence of rotations, reflections, and translations that take one figure to another, then they are congruent.

For the second part about drawing \(D'\):

1. First, determine the type of rigid - transformation(s) (rotation, reflection, translation) that map \(\triangle ABC\) to \(\triangle A'B'C'\).

• If it is a translation, measure the displacement vector from \(A\) to \(A'\), \(B\) to \(B'\), and \(C\) to \(C'\). Then apply the same displacement vector to point \(D\) to get \(D'\).
• If it is a rotation, find the center of rotation, the angle of rotation. Rotate point \(D\) around the center of rotation by the same angle to get \(D'\).
• If it is a reflection, find the line of reflection. Reflect point \(D\) across the line of reflection to get \(D'\).
• In general, if it is a combination of these transformations, apply them one - by - one to point \(D\) in the same order as they are applied to \(\triangle ABC\) to get \(D'\). Then label the new point as \(D'\) on the diagram. But without specific information about the transformation in the given diagram, we can only provide the general method.