Exercise 31 Page 206

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 $\overline{K K^{''}}$ 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 $\overline{K K^{''}}$ is perpendicular to both line $ℓ$ and line $m .$ Let's show this as a two-column proof.

Statement	Reason
A reflection in line $ℓ$ maps $\overline{J K}$ to $\overline{J^{'} K^{'}},$ a reflection in line $m$ maps $\overline{J^{'} K^{'}}$ to $\overline{J^{''} K^{''}},$ and $ℓ ∥ m$	Given
If $\overline{K K^{''}}$ intersects line $ℓ$ at $L$ and line $m$ at $M,$ then $L$ is the perpendicular bisector of $\overline{K K^{'}},$ and $M$ is the perpendicular bisector of $\overline{K^{'} K^{''}}$	Definition of reflection
$\overline{K K^{'}}$ is perpendicular to $ℓ$ and $m,$ and $K L = L K^{'}$ and $K^{'} M = M K^{''}$	Definition of perpendicular bisector

b From Part

A

we know that

K L = L K^{'}

and

K^{'} M = M K^{''} .

Let's call the length of these segments

a

and

b

respectively and also show the lengths

K K^{''}

and

d .

Segment Addition Postulate
If $B$ is between $A$ and $C,$ then $A B + B C = A C .$

Using this postulate, we can write the following two equations.

K K^{''} d = a + a + b + b = a + b

Now we can show that

K K^{''} = 2 d .

K K^{''} = a + a + b + b

Add terms

K K^{''} = 2 (a + b)

$(a + b) = d$

K K^{''} = 2 d

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

Statement	Reason
$K L = L K^{'}, K^{'} M = M K^{''}$	Given
If $d$ is the distance between $ℓ$ and $m,$ then $d = L M$	Ruler Postulate
$L M = L K^{'} + K^{'} M K K^{''} = K L + L K^{'} + K^{'} M + M K^{''}$	Segment Addition Postulate
$K K^{''} = L K^{'} + L K^{'} + K^{'} M + K^{'} M$	Substitution Property of Equality
$K K^{''} = 2 (L K^{'} + K^{'} M)$	Distributive Property
$K K^{''} = 2 (L M)$	Substitution Property of Equality
$K K^{''} = 2 d$	Transitive Property of Equality

Communicate Your Answer p.199

Monitoring Progress p.200-203