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Basic Geometrical Ideas - Intersecting and Parallel Lines

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Definition of a Line: A line is a straight path that extends infinitely in both directions. It is represented by AB↔\overleftrightarrow{AB} and has no thickness. Visually, a line is drawn with arrows on both ends to show it never stops.

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Intersecting Lines: Two lines are said to be intersecting if they have exactly one point in common. This common point is called the point of intersection. Visually, these lines look like the letter XX or a crossroad where two paths meet at a single spot PP.

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Parallel Lines: Lines in a plane that do not meet or intersect at any point, no matter how far they are extended, are called parallel lines. Examples include the opposite edges of a ruler or railway tracks. We denote that line ll is parallel to line mm as l∥ml \parallel m.

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Distance between Parallel Lines: The perpendicular distance between two parallel lines remains constant throughout their length. If the distance is dd at one point, it will be dd at every other point along the lines.

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Concurrent Lines: If three or more lines in a plane pass through the same point, they are called concurrent lines. This resembles a starburst pattern or the spokes of a bicycle wheel meeting at the central hub.

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Real-world Examples: Intersecting lines can be seen in the English letter XX or the letter YY. Parallel lines can be seen in the two vertical bars of the letter HH or the horizontal lines on a page of a notebook.

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Point of Intersection: The specific location where two intersecting lines cross is a single point. If line ll and line mm intersect at point OO, then point OO lies on both lines simultaneously.

📐Formulae

Notation for a line passing through points AA and BB: AB↔\overleftrightarrow{AB}

Symbolic representation of parallel lines: l∥ml \parallel m

Intersection of two lines ll and mm at point PP: l∩m={P}l \cap m = \{P\}

Condition for Parallelism: If the distance between two lines l1l_1 and l2l_2 is dd at any point xx, and dd is constant for all xx, then l1∥l2l_1 \parallel l_2.

💡Examples

Problem 1:

Look at a standard window frame with a cross-grid. Identify the types of lines formed by the wooden bars and name the relationship between the top horizontal bar and the bottom horizontal bar.

Solution:

  1. The vertical bars and horizontal bars meet at specific points, so they are intersecting lines.
  2. The top horizontal bar and the bottom horizontal bar run in the same direction and never meet, so they are parallel lines.
  3. Symbolically, if the top bar is L1L_1 and the bottom is L2L_2, we write L1∥L2L_1 \parallel L_2.

Explanation:

The problem applies the definitions of intersection (meeting at a point) and parallelism (never meeting) to a real-world object.

Problem 2:

If line ll intersects line mm at point AA, and line nn also passes through point AA, what is the special name given to these three lines?

Solution:

  1. Line ll and line mm share point AA.
  2. Line nn also shares point AA.
  3. Since three lines (l,m,nl, m, n) all pass through the same common point AA, they are called concurrent lines.

Explanation:

By definition, when three or more lines intersect at the same single point, they are classified as concurrent.