Question

question 15 (multiple choice worth 1 points) (03 02r mc) the abc toy company is creating two similar pieces for a board game, as shown. how could the toy maker confirm that the pieces are similar using a transformation? verify ∠a ≅ ∠d and ∠b ≅ ∠e by translation. verify ab ≅ de and bc ≅ ef by translation. verify ∠a ≅ ∠d and ∠b ≅ ∠e by reflection. verify ab ≅ de and bc ≅ ef by reflection

Explanation:

Step1: Recall the properties of similar triangles

Similar triangles have corresponding angles equal. Transformations like translation, rotation, reflection don't change the shape and size of the figure (for congruent figures) but for similar figures (where one is a scaled - version of the other), we can use similarity transformations. Here, we are just checking for similarity (angle - angle similarity criterion can be used).

Step2: Analyze the options

• Option 1: Verifying \(\angle A\cong\angle D\) and \(\angle B\cong\angle E\) by translation. Translation is a rigid transformation. If we can map one triangle to another (after possible dilation) such that corresponding angles are equal (by translation which preserves angle measures), it helps in confirming similarity.
• Option 2: Verifying \(\overline{AB}\cong\overline{DE}\) and \(\overline{BC}\cong\overline{EF}\) by translation. Congruent sides (by translation, a rigid transformation) would imply congruent triangles (not just similar) if all corresponding sides are congruent. But we are looking for similarity.
• Option 3: Verifying \(\angle A\cong\angle D\) and \(\angle B\cong\angle E\) by reflection. Reflection is a rigid transformation. Just like translation, if we can show that corresponding angles are equal (angle - angle similarity: two pairs of corresponding angles equal implies similarity).
• Option 4: Verifying \(\overline{AB}\cong\overline{DE}\) and \(\overline{BC}\cong\overline{EF}\) by reflection. Congruent sides (by reflection, a rigid transformation) would imply congruent triangles (not just similar) if all corresponding sides are congruent.

Since we know that for two triangles \(\triangle ABC\) and \(\triangle DEF\), if \(\angle A=\angle D\) and \(\angle B = \angle E\), then \(\triangle ABC\sim\triangle DEF\) (by AA - angle - angle similarity criterion). And translation (a type of transformation that slides a figure) can be used to show the correspondence of angles.

Answer:

Verify \(\angle A\cong\angle D\) and \(\angle B\cong\angle E\) by translation.