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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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Intersecting Lines: Two distinct lines are called intersecting lines if they have a common point. This common point is called the point of intersection. For example, the letter 'X' is formed by two intersecting line segments.

Two lines l and m intersecting at a single point P.
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Parallel Lines: Lines in a plane that do not meet, however far they are extended, are called parallel lines. The distance between them remains constant throughout. Examples include the tracks of a railway line or the opposite edges of a ruler.

Two horizontal lines p and q that never meet.
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Concurrent Lines: When three or more lines in a plane pass through the same point, they are called concurrent lines, and that point is the point of concurrence.

Three lines passing through a common center point O.
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Relationship in Shapes: In polygons like rectangles, adjacent sides are intersecting (meeting at the vertex), while opposite sides are parallel.

📐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.

Problem 3:

In the given figure of a rectangle ABCDABCD, identify all pairs of parallel line segments and any two pairs of intersecting line segments.

A rectangle ABCD showing parallel and intersecting sides.

Solution:

Parallel pairs: AB∥DCAB \parallel DC and AD∥BCAD \parallel BC. Intersecting pairs: (AB,BC)(AB, BC) at point BB, and (AD,DC)(AD, DC) at point DD.

Explanation:

In a rectangle, opposite sides never meet even if extended, hence they are parallel. Adjacent sides meet at the corners (vertices), making them intersecting lines.

Problem 4:

Observe the lines l1l_1, l2l_2, and l3l_3. If l1l_1 is parallel to l2l_2, and l3l_3 cuts across both, identify the points of intersection.

Line l3 intersecting parallel lines l1 and l2 at points P and Q.

Solution:

The points of intersection are PP (where l3l_3 meets l1l_1) and QQ (where l3l_3 meets l2l_2).

Explanation:

A line that intersects two or more lines is often called a transversal. Here, l3l_3 creates two distinct intersection points because l1l_1 and l2l_2 are separated by a constant distance.