State giving reasons, whether the following statements are true or false : In all the following questions the line does not contain a side of the triangle.
(1) A line can be drawn in the plane of a triangle not intersecting any of the sides of a triangle.
(2) A line can be drawn in the plane of a triangle which is not passing through any of the three vertices and intersecting all the three sides of the triangle.
(3) If a line drawn in the plane of a triangle intersects the triangle at only one point, the line passes through a vertex of the triangle.
(4) If a line intersects two of the three sides of a triangle in two distinct points and does not intersect the third side, then the line is parallel to the third side.
(5)In the plane of ΔABC, a line I can be drawn such that l ∩ BC = {P},l ∩ AC = {0} and l ∩ AB = ∅.
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(1) The given statement is true because for a line in the plane of a Δ, there are 3 possibilities
i) The line does not intersect the Δ
ii) The line intersects the Δ at one point
iii) The line intersects the Δ at two points
The first possibility is satisfied; hence the given statement is true. [see figure ]
(2) The given statement is false because according to the theorem, if a line lying in the plane of a Δ and not passing through any vertex intersects one side, then it can intersect maximum two sides of ∆. [see figure]
(3) The given statement is true because if a line passing through a vertex of ∆, then it intersects ∆ only one point.[see figure]
(4) The given statement is false because if a line intersects two sides of a Δ and does not intersect the third side, it can intersect the line containing the third side.
(5) The given statement is true because in the plane of ΔABC, if line l intersects BC and AC at distinct points P and Q respectively, then according to the theorem, if a line lying in the plane of a Δ and not passing through any vertex intersects one side, then it does intersect one more side but does not intersect the third side, it cannot intersect AB.
i) The line does not intersect the Δ
ii) The line intersects the Δ at one point
iii) The line intersects the Δ at two points
The first possibility is satisfied; hence the given statement is true. [see figure ]
(2) The given statement is false because according to the theorem, if a line lying in the plane of a Δ and not passing through any vertex intersects one side, then it can intersect maximum two sides of ∆. [see figure]
(3) The given statement is true because if a line passing through a vertex of ∆, then it intersects ∆ only one point.[see figure]
(4) The given statement is false because if a line intersects two sides of a Δ and does not intersect the third side, it can intersect the line containing the third side.
(5) The given statement is true because in the plane of ΔABC, if line l intersects BC and AC at distinct points P and Q respectively, then according to the theorem, if a line lying in the plane of a Δ and not passing through any vertex intersects one side, then it does intersect one more side but does not intersect the third side, it cannot intersect AB.
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