Study the following statement: “Two intersecting lines cannot be perpendicular to the same line”.
Check whether it is an equivalent version to the Euclid’s fifth postulate. [Hint: Identify the two intersecting lines $l$ and $m$ and the line $n$ in the above statement.]

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Solution

Verify the given statement:
Given statement is “Two intersecting lines cannot be perpendicular to the same line”.
According to Euclid's fifth postulate, a straight line intersecting two straight lines produces internal angles on the same side that, when added together, are less than $180 °$ , if two straight lines are generated endlessly, they will eventually meet or intersect on the side where the total number of angles is fewer than $180 °$ .
Let the lines $l$ and $m$ are perpendicular on line $n$
Then lines $l$ and $m$ become parallel and thus never intersect.
However, if two straight lines are generated, they will eventually meet or intersect, according to Euclid's fifth postulate.
If the lines $l$ and $m$ are perpendicular on line $n$ can never intersect thus contradicts the Euclid’s fifth postulate.
$\Rightarrow$ To be in the equivalent version of Euclid’s fifth postulate the lines $l$ and $m$ must not be perpendicular on same line.
Hence, the given statement “Two intersecting lines cannot be perpendicular to the same line” is an equivalent version to the Euclid’s fifth postulate

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Similar questions

Q. State which of the following statements are true (T) or which are false (F):

(i) If two lines in the same plane do not interest, then they much be parallel.
(ii) Distance between two parallel lines is not same every where.
(iii) If m ⊥ i, n ⊥ i and m ≠ n, then m ∥ n.
(iv) Two non-intersection coplanar rays are parallel.
(v) If ray AB ∥ line , then line segment AB ∥ m.
(vi) If line AB ∥ line , then line segment AB ∥ m.
(vii) No two parallel segments intersect.
(viii) Every pair of lines is a pair of coplanar lines.
(ix) Two lines perpendicular to the same line are parallel.
(x) A line perpendicular to one of two parallel lines is perpendicular to the other.

Question 4 (c)

Study the diagram. The line $l$ is perpendicular to line $m$ .

Identify any two line segments for which PE is the perpendicular bisector.

Which of the following statements are true (I) and which are false (F)? Give reasons.
(i) If two lines are intersected by a transversal, then corresponding angles are equal.
(ii) If two parallel lines are intersected by a transversal, then alternate interior angles are equal.
(iii) Two lines perpendicular to the same line are perpendicular to each other.
(iv) Two lines parallel to the same line are parallel to each other.
(v) If two parallel lines are intersected by a transversal, then the interior angles on the same side of the transversal are equal.

Q. Study the following statement:
“Two intersecting lines cannot be perpendicular to the same line.”Check whether it is an equivalent version to the Euclid’s fifth postulate.

Q. Read the following statement carefully and identify the true statement
(a) Two lines parallel to a third line are parallel
(b) Two lines perpendicular to a third line are parallel
(c) Two lines parallel to a plane are parallel
(d) Two lines perpendicular to a plane are parallel
(e) Two lines either intersect or are parallel

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Study the following statement: “Two intersecting lines cannot be perpendicular to the same line”. Check whether it is an equivalent version to the Euclid’s fifth postulate. [Hint: Identify the two intersecting lines l and m and the line n in the above statement.]

Study the following statement: “Two intersecting lines cannot be perpendicular to the same line”.
Check whether it is an equivalent version to the Euclid’s fifth postulate. [Hint: Identify the two intersecting lines $l$ and $m$ and the line $n$ in the above statement.]