Exercise 31 Page 206

Practice makes perfect

Reflections in Parallel Lines Theorem
If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is the same as a translation.

Next, we will draw KK'' and name the intersection with l and m accordingly.

According to the definition of a reflection, L is on the perpendicular bisector of line l, and M is on the perpendicular bisector of line m.

Now we have proven that KK'' is perpendicular to both line l and line m. Let's show this as a two-column proof.

Statement	Reason
1. A reflection in line l maps JK to J'K', a reflection in line m maps J'K' to J''K'', and l ∥ m	1. Given
2. If KK'' intersects line l at L and line m at M, then L is the perpendicular bisector of KK', and M is the perpendicular bisector of K'K''	2. Definition of reflection
3. KK' is perpendicular to l and m, and KL=LK' and K'M=MK''	3. Definition of perpendicular bisector

b From Part A we know that KL=LK' and K'M=MK''. Let's call the length of these segments a and b respectively and also show the lengths KK'' and d.

Segment Addition Postulate
If B is between A and C, then AB+BC=AC.

Using this postulate, we can write the following two equations. KK''&=a+a+b+b d&=a+b Now we can show that KK''=2d.

KK''=a+a+b+b

Add terms

KK''=2(a+b)

(a+b)= d

KK''=2 d

Let's show this as a two-column proof.

Statement	Reason
1. KL=LK', K'M=MK''	1. Given
2. If d is the distance between l and m, then d=LM	2. Ruler Postulate
3. &LM=LK'+K'M &KK''=KL+LK'+K'M+MK''	3. Segment Addition Postulate
4. KK''=LK'+LK'+K'M+K'M	4. Substitution Property of Equality
5. KK''=2(LK'+K'M)	5. Distributive Property
6. KK''=2(LM)	6. Substitution Property of Equality
7. KK''=2d	7. Transitive Property of Equality

Communicate Your Answer p.199

Monitoring Progress p.200-203

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