Determine whether each statement is true or false in double-struck r3. (a) two lines parallel to a third line are parallel. true false (b) two lines perpendicular to a third line are parallel. true false (c) two planes parallel to a third plane are parallel. true false (d) two planes perpendicular to a third plane are parallel. true false (e) two lines parallel to a plane are parallel. true false (f) two lines perpendicular to a plane are parallel. true false (g) two planes parallel to a line are parallel. true false (h) two planes perpendicular to a line are parallel. true false (i) two planes either intersect or are parallel. true false (j) two lines either intersect or are parallel. true false (k) a plane and a line either intersect or are parallel. true false

This question is based on the properties of lines and planes. Therefore, (a) True , (b) False , (c) True , (d) False , (e) False , (f) True , (g) False

, (h) True , (i) True , (j) False , (k) True.

We have to choose correct statement and marked true or false.

Lets solve the problem.

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

Reason - The direction vectors are scalar multiple of the direction of the third  line, then they are scalar multiple of each other. So, they are parallel.

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

Reason- As we know that, x-axis and the y-axis are perpendicular to the z-axis but not parallel to  each other.

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

Reason- The normal vectors of planes are parallel to the normal vector of the third  plane. So, these two normal vectors are parallel to each other and the  planes are parallel.

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

Reason- x-y plane and y-z plane are not parallel to each other. But they are perpendicular to x-z plane.

(e) Two lines parallel to a plane are parallel. (False)

Reason - Both  x-axis and y-axis are not parallel to each other. But, parallel  to the plane z = 1.

(f) Two lines perpendicular to a plane are parallel. (True)

Reason - The direction vectors of the lines parallel to the normal vector of  the plane, then they parallel to each other , so the lines are parallel.

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

Reason- The planes y = 1 and z = 1 are not parallel, but they are parallel to the x-axis.

(h) Two planes perpendicular to a line are parallel. (True)

Reason- The normal vectors of the two planes are parallel to the direction of  line. So, they are parallel to each other. Hence, they are parallel.

(i) Two planes either intersect or are parallel (True)

(j) Two lines either intersect or are parallel (False)

Reason- They can also be  skew.

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

Reason- They are parallel, if the normal vector of the plane and the direction  of the line are perpendicular to each other, otherwise the line intersect the plane  at the angle 90° .

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(a) True , (b) False , (c) True , (d) False , (e) False , (f) True , (g) False

(h) True , (i) True , (j) False , (k) True

Step-by-step explanation:

* Lets explain how to solve the problem

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

- Their direction vectors are scalar multiplies of the direction of the 3rd

 line, then they are scalar multiples of each other so they are parallel

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

- The x-axis and the y-axis are ⊥ to the z-axis but not parallel to

  each other

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

- Their normal vectors parallel to the  normal vector of the 3rd plane,

  so these two normal vectors are parallel to each other and the

  planes are parallel

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

- The xy plane and yz plane are not parallel to each other but both

 ⊥ to xz plane

(e) Two lines parallel to a plane are parallel (False)

- The x-axis and y-axis are not parallel to each other but both parallel

  to the plane z = 1

(f) Two lines perpendicular to a plane are parallel (True)

- The direction vectors of the lines parallel to the normal vector of

  the plane, then they parallel to each other , so the lines are parallel

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

- The planes y = 1 and z = 1 are not parallel but both are parallel to

  the x-axis

(h) Two planes perpendicular to a line are parallel (True)

- The normal vectors of the 2 planes are parallel to the direction of

  line, then they are parallel to each other so the planes are parallel

(i) Two planes either intersect or are parallel (True)

(j) Two lines either intersect or are parallel (False)

- They can be skew

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

- They are parallel if the normal vector of the plane and the direction

  of the line are ⊥ to each other , otherwise the line intersect the plane

  at the angle 90° - Ф

the answer is a i belaeve atleast thats what my calculator just gfave me

step-by-step explanation: