2. For a translation, is the measure of the angle between two rays an invariant?

Ans: Yes.

4.Does the set of all reflections about lines in a plane include an identity?

Ans:No.

6. Explain how a segment and its image might be parallel for a rotation other than the identity.

Solution:The angle of rotation could be 180º .

8. Show that a reflection is an involutory transformation.

Solution: If the image of a point P under a reflection is P', then the image of P' under the same reflection is P. Thus, if R is the reflection, then RR = I, the identity. Therefore, R is involutory.

10. Given an example in which a segment and its image are parallel under reflection.

Solution: The segment is parallel or perpendicular to the line of reflection.

12. For a glide reflection, show that the image point will be the same if the translation is followed by the reflection.

Solution: Let P be a point and l the line of reflection of the glide reflection (ignore the special case of P being on l). Let P' be the image of P under the reflection about l, and P" be the image of P' under the translation in the direction parallel to l. Now, consider P''', the image of P under this same translation, and Q the image of P''' under the reflection about l. We wish to show that Q = P". The line segments PP' and P'''Q are parallel since they are both perpendicular to l. The image of the line segment PP' under the translation is parallel to PP' (a property of translations) and passes through P''' (the image of P). Since there is only one parallel to a given line through a given point not on that line, P'''Q must be the image of PP' under the translation. This means that P" (the image of P' under the translation) lies on the line P'''Q. The line segment PP''' is parallel to l (since the translation is in a direction parallel to l) and so is the line segment P'P" (for the same reason). The image of PP''' under the reflection is parallel to l (not a general property of reflections, but true in this case because the line starts out parallel to the line of reflection). This image is P'Q, and so, is a line through P' which is parallel to l. Again, because only one line can be parallel to a given line through a point not on that line, P'Q must be P'P", and this forces Q = P".

14. When are a segment and its image parallel under a glide reflection?

Solution: When the segment is parallel or perpendicular to the line of reflection.

For exercises 16 and 18 use the figure below and sketch the image of triangle ABC under each.

16. A translation represented by the vector shown.

Solution:

18. A reflection about the line shown.

Solution:

For exercises 20 and 22 use the figure below and sketch the image of triangle ABC under each.

20. A translation represented by the vector shown.

Solution:

22. A reflection about the line shown.

Solution:

24. Sketch, if possible, an example of each of the following. If no example exists, give a reason.

1. Two congruent rectangles such that one can be mapped to the other by both a rotation and a translation.
2. Two congruent squares such that one cannot be mapped to the other by a translation.
3. Two congruent circles such that one cannot be mapped to the other by a reflection or a translation.

Solution:

(a)

(b)

(c) No possible example, two congruent circles can always be mapped to each other by a translation that sends the center of one to the center of the other.

25. Which specific transformation could be used in a proof of each of the following?

1. For an isosceles triangle, the medians to the two congruent sides are congruent.
2. If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
3. The diagonals of a rhombus are perpendicular.
4. Each diagonal forms two congruent triangles with the sides of the parallelogram.
5. If two circles intersect in two points, then their common chord is perpendicular to the line determined by their centers.

Solution:

1. Reflection about the altitude to the base.
2. Translation along the transversal.
3. Reflection about a diagonal.
4. Rotation of 180º about point of intersection of the diagonals.
5. Reflection about the line determined by the centers.