krit.club logo

Practical Geometry - Construction of a Line Parallel to a Given Line

Grade 7CBSE

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

🔑Concepts

•

Parallel lines are lines in a plane that never meet or intersect, no matter how far they are extended in either direction. Visually, they look like the opposite edges of a straight ruler or railway tracks, maintaining a constant perpendicular distance between them.

•

A transversal is a line that intersects two or more lines at distinct points. In the construction of parallel lines, the transversal is used to create specific angle relationships between the given line and the new line being constructed.

•

The Property of Alternate Interior Angles is the primary logic used in this construction. It states that if a transversal intersects two lines such that a pair of alternate interior angles are equal, then the lines must be parallel. Visually, these angles form a 'Z' or 'N' shape pattern across the transversal.

•

To construct a parallel line, we start with a line ll and a point PP not on it. We draw a transversal line passing through PP and any point XX on line ll. This creates an angle at the intersection point XX, which we then replicate at point PP.

•

The Corresponding Angles Property is another geometric principle where, if a transversal cuts two lines such that the corresponding angles (angles in the same relative position at each intersection) are equal, the lines are parallel. This is visually represented by an 'F' shape pattern.

•

Geometric construction requires precision using a ruler and a compass. Unlike freehand drawing, a compass ensures that arc lengths and angle measurements are perfectly transferred from one point to another, ensuring the lines are mathematically parallel.

•

The distance between two parallel lines is constant throughout. Visually, if you draw a perpendicular line from any point on one parallel line to the other, the length of that perpendicular segment will always be the same.

📐Formulae

If ∠1=∠2\angle 1 = \angle 2 (Alternate Interior Angles), then l∥ml \parallel m

If ∠1=∠3\angle 1 = \angle 3 (Corresponding Angles), then l∥ml \parallel m

Sum of interior angles on the same side of transversal: ∠1+∠4=180∘\angle 1 + \angle 4 = 180^\circ (Co-interior angles)

Distance dd between lines l1l_1 and l2l_2 is constant: d(P,l2)=d(Q,l2)d(P, l_2) = d(Q, l_2) for any points P,QP, Q on l1l_1

💡Examples

Problem 1:

Draw a line XYXY. Take a point AA outside it. Through AA, draw a line mm parallel to XYXY using the concept of alternate interior angles.

Solution:

  1. Draw a line XYXY and mark a point AA outside the line.
  2. Mark any point BB on the line XYXY and join the points AA and BB. Now, ABAB is the transversal.
  3. With BB as the center and any convenient radius, draw an arc cutting XYXY at point CC and ABAB at point DD.
  4. With AA as the center and the same radius as in step 3, draw an arc EFEF cutting ABAB at point GG.
  5. Place the compass pointer at CC and adjust the opening to measure the distance to DD.
  6. With the same opening and GG as the center, draw an arc to cut the arc EFEF at point HH.
  7. Draw a line mm passing through points AA and HH.

Explanation:

This construction replicates the angle ∠ABC\angle ABC at point AA such that ∠BAH=∠ABC\angle BAH = \angle ABC. Since these are alternate interior angles and are made equal, line mm becomes parallel to line XYXY.

Problem 2:

Given a line ll and a point PP at a distance of 44 cm from it, construct a line mm parallel to ll passing through PP.

Solution:

  1. Draw a line ll.
  2. Take any point XX on line ll and draw a perpendicular line XYXY using a protractor or compass at 90∘90^\circ.
  3. With XX as the center and a radius of 44 cm on the compass, draw an arc cutting the perpendicular line at point PP.
  4. At point PP, draw another perpendicular line to the segment XPXP.
  5. Extend this line on both sides to name it line mm.

Explanation:

Since line mm is perpendicular to XPXP and line ll is also perpendicular to XPXP, line mm and line ll are parallel because they are both perpendicular to the same transversal line at a distance of 44 cm.