Question

Determine which statements are true in the set of real numbers3. (Select all that apply.) (a) Two lines parallel to a third line are parallel. (b) Two lines perpendicular to a third line are parallel. (c) Two planes parallel to a third plane are parallel. (d) Two planes perpendicular to a third plane are parallel. (e) Two lines parallel to a plane are parallel. (f) Two lines perpendicular to a plane are parallel. (g) Two planes parallel to a line are parallel. (h) Two planes perpendicular to a line are parallel. (i) Two planes either intersect or are parallel. (j) Two lines either intersect or are parallel. (k) A plane and a line either intersect or are parallel. Incorrect: Your answer is incorrect.

Answer 1

In the set of real numbers, the true statements are: (a) Two lines parallel to a third line are parallel. (c) Two planes parallel to a third plane are parallel. (g) Two planes parallel to a line are parallel. (i) Two planes either intersect or are parallel. (j) Two lines either intersect or are parallel.

Step-by-step explanation:

The correct statements in the set of real numbers are:

(a) Two lines parallel to a third line are parallel.

(c) Two planes parallel to a third plane are parallel.

(g) Two planes parallel to a line are parallel.

(i) Two planes either intersect or are parallel.

(j) Two lines either intersect or are parallel.

To determine which statements are true in the set of real numbers, you need to understand the concepts of parallel lines and planes. In general, two lines are parallel if they have the same slope and will never intersect. Similarly, two planes are parallel if they do not intersect and have the same normal vector. Using these concepts, you can determine which statements are true by checking if the given lines or planes satisfy the conditions for being parallel.

Answer 2

(a) True

(b) False

(c) True

(d) False

(e) False

(f) True

(g) False

(h) True

(i) True

(k) True

Step-by-step explanation:

(a) Two lines parallel to a third line are parallel

True

(b) Two lines perpendicular to a third line are parallel

Only for lines on the same plane

Therefore, false

(c) Two planes parallel to a third plane are parallel

True

(d) Two planes perpendicular to a third plane are parallel

The two planes can be at an angle to each other and so intersect

Therefore, false

(e) Two lines parallel to a plane are parallel

Where the two lines are on a plane parallel to the first plane but the lines are not themselves parallel to each other they intersect

Therefore, false

(f) Two lines perpendicular to a plane are parallel

True

(g) Two planes parallel to a line are parallel

Where the planes are not parallel to each other, they will intersect

Therefore, false

(h) Two planes perpendicular to a line are parallel

True

(i) Two planes either intersect or are parallel

True

(k) A plane and a line either intersect or are parallel

True.