Chapter 6, Problem 1RP

a

To determine

To check whether the statement “Two lines must either intersect or be parallel” is true or false.

Answer to Problem 1RP

True

Explanation of Solution

Given information :

The following statement is given

“Two lines must either intersect or be parallel”

If a line is parallel, then it will never intersect each other. Therefore, the intersecting lines could not be parallel.

Hence the statement is correct that either the line can be intersecting or parallel.

b

To determine

To indicate whether the statement “In a plane, two line perpendicular to same line are parallel” is true or false

Answer to Problem 1RP

True

Explanation of Solution

Given information :

The following statement is given

“Two line perpendicular to same line are parallel”

In a plane, the two lines perpendicular to the same line would be parallel to each other as shown in the figure.

  

c

To determine

The statement “In a space, two lines perpendicular to same line are parallel” is true or false.

Answer to Problem 1RP

False

Explanation of Solution

Given information :

The following statement is given

“In a space, two lines perpendicular to same line are parallel”

Let us consider two non-parallel lines in a vertical plane, these both lines must be perpendicular to the horizontal plane.

Therefore, the two lines need not to be parallel, if they are perpendicular to the same line.

d

To determine

To check whether the given statement “if a line is perpendicular to plane, it is perpendicular to every line in the plane” is true or false

Answer to Problem 1RP

True

Explanation of Solution

Given information :

The following statement is given:

“If a line is perpendicular to plane, it is perpendicular to every line in the plane”

If a line is perpendicular to the plane, that means it will remain perpendicular to the every line in the other plane.

  

As we can see from the above figure, any line in the vertical plane must be perpendicular to the every line in the horizontal plane.

e

To determine

The statement “Is it possible for two planes to intersect at one point” is true or false.

Answer to Problem 1RP

False

Explanation of Solution

Given information :

“Is it possible for two planes to intersect at one point”

Explanation:

The planes are containing infinite no. of points, so when intersection of the planes takes place, then it intersects at infinite no. of points, therefore it can not be possible to intersect at one point.

f

To determine

The following statement “If a line is perpendicular to a line in a plane, it is perpendicular to plane” is true or false.

Answer to Problem 1RP

False

Explanation of Solution

Given information :

“If a line is perpendicular to a line in a plane, it is perpendicular to plane”

When line is perpendicular to the other line in the same plane, then that line can not be perpendicular to the plane. Because to be perpendicular to the plane that line must be in some other plane.

g

To determine

The following statement “Two lines perpendicular to same line are parallel” is true or false

Answer to Problem 1RP

False

Explanation of Solution

Given information :

“Two lines perpendicular to same line are parallel”

Explanation:

If all the lines are in the same plane, then we can say that the line which are perpendicular to the same line would be parallel.

But if the lines are not in the same plane, then it is not need to be true.

h

To determine

The statement “A triangle is plane figure” is true or false

Answer to Problem 1RP

True

Explanation of Solution

Given information :

“ A triangle is plane figure”

A triangle is the 2D figure that means it has to be in a plane. If we go for 3D, then it becomes pyramid. Hence, we can say that the triangle is a plane figure.

i

To determine

The following statement “A line that is perpendicular to horizontal is vertical” is true or false

Answer to Problem 1RP

True

Explanation of Solution

Given information :

“ A line that is perpendicular to horizontal is vertical”

If a line is perpendicular to the horizontal line, then it has to be vertical. Otherwise it will not be at ${90}^o$ or perpendicular to the horizontal line.

j.

To determine

The following statement “Three parallel lines must be coplanar” is true or false.

Answer to Problem 1RP

False

Explanation of Solution

Given information :

“ Three parallel lines must be coplanar”

It can be possible that three lines are in two different planes and they can be parallel to each other. Therefore, three lines need not to be in the same plane for becoming parallel.

k.

To determine

The statement “Every four-sided figure is plane figure” is true or false.

Answer to Problem 1RP

True

Explanation of Solution

Given information :

“Every four-sided figure is plane figure”

Plane figure means that all figure is 2D or all the sides are in the same plane. Therefore, if we increase the plane, then four-sided figure would not be possible. Therefore, we can say every four-sided figure would be a plane figure.