Chapter 5.1, Problem 37WE

a)

To determine

To find: If line segments $AX$ and $BY$ are equal.

Answer to Problem 37WE

Thus, it proves that

  $AX=BY$ .

Explanation of Solution

Given information: Plane P and plane Q are parallel to each other as well as two lines j and k passing from two planes are also parallel as shown in below figure,

  

Concept used: Following concept of theorem 5.4 is used here, “The opposite sides of a parallelogram are congruent.”

Calculation: As plane P and plane Q are parallel, so the line segments lying on them will also be parallel to each other, so, $AB\left|\right|XY$ .

Also, as lines j and k are parallel, so their corresponding line segments $AX$ and $BY$ will also be parallel. And hence $ABYX$ is a parallelogram. Accordingly, its opposite sides $AX$ and $BY$ will be equal to each other.

Conclusion: Thus, it is concluded that $AX=BY$ .

b)

To determine

To find: The theorem about parallel planes and lines that are used to prove $AX=BY$ .

Answer to Problem 37WE

Theorem 3.1 that is “If two parallel planes are cut by a third plane, then the lines of intersection are parallel.” Is used here.

Explanation of Solution

Given information: Plane P and plane Q are parallel to each other as well as two lines j and k passing from two planes are also parallel as shown in below figure,

  

Concept used: Following concept of theorem 3.1 to be used here, ” If two parallel planes are cut by a third plane, then the lines of intersection are parallel.”

Calculation: By using the concept of theorem 3.1 that says that If two parallel planes are cut by a third plane, then the lines of intersection are parallel.” , it is proved here that $ABYX$ is a parallelogram. Accordingly, its opposite sides $AX$ and $BY$ will be equal to each other.

Conclusion: So, statement of theorem 3.1 is used here to prove $AX=BY$ .