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Parallel and Intersecting Lines - Drawing Parallel Lines

Grade 7CBSE

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

🔑Concepts

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Parallel lines are lines in a plane that do not intersect or meet at any point, no matter how far they are extended. The distance between them remains constant throughout.

Two parallel lines l and m with a constant distance d between them.
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To draw a line parallel to a given line ll through a point PP not on it, we can use the property of alternate interior angles. By drawing a transversal through PP and creating an equal alternate angle at PP, we ensure the lines are parallel.

Construction of a parallel line using equal alternate interior angles.
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The 'Perpendicular Method' involves drawing a perpendicular segment of a specific length hh from line ll to a point XX, and then drawing another perpendicular to that segment at point XX. This second line will be parallel to ll at distance hh.

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When constructing parallel lines, we use a ruler and a compass to copy an angle. If the corresponding angles created by a transversal are made equal, the two lines must be parallel.

Visualizing equal corresponding angles to verify parallelism.

📐Formulae

If l∥m and t is a transversal, then alternate interior angles are equal: ∠1=∠2\text{If } l \parallel m \text{ and } t \text{ is a transversal, then alternate interior angles are equal: } \angle 1 = \angle 2

If l∥m, then corresponding angles are equal: ∠1=∠5\text{If } l \parallel m, \text{ then corresponding angles are equal: } \angle 1 = \angle 5

Sum of co-interior angles=180∘\text{Sum of co-interior angles} = 180^\circ

Condition for parallelism: ∠Alternate Interior1=∠Alternate Interior2  ⟹  l∥m\text{Condition for parallelism: } \angle \text{Alternate Interior}_1 = \angle \text{Alternate Interior}_2 \implies l \parallel m

💡Examples

Problem 1:

Draw a line ll. Draw a perpendicular to ll at any point on ll. On this perpendicular, choose a point XX, 4 cm4 \text{ cm} away from ll. Through XX, draw a line mm parallel to ll.

Solution:

  1. Draw line ll. 2. Mark a point PP on ll. 3. Construct a perpendicular line at PP using a protractor or compass. 4. Measure 4 cm4 \text{ cm} from PP on this perpendicular and mark it as XX. 5. At XX, construct another perpendicular to the line PXPX. 6. Name this new line mm. Then m∥lm \parallel l.

Explanation:

Since the alternate interior angles or the corresponding angles (both 90∘90^\circ here) are equal, the lines ll and mm are parallel. The distance between them is constant at 4 cm4 \text{ cm}.

Problem 2:

If a transversal tt intersects two lines ll and mm such that a pair of alternate interior angles are 65∘65^\circ and xx, what must be the value of xx for l∥ml \parallel m?

Solution:

x=65∘x = 65^\circ

Explanation:

According to the property of parallel lines, if a transversal intersects two parallel lines, the alternate interior angles must be equal. Therefore, for ll to be parallel to mm, xx must be equal to 65∘65^\circ.

Problem 3:

Draw a line segment XY=6 cmXY = 6\text{ cm}. Take a point ZZ outside it. Construct a line mm passing through ZZ and parallel to XYXY using the concept of alternate interior angles.

Construction of line m parallel to XY through point Z.

Solution:

  1. Draw XY=6 cmXY = 6\text{ cm} using a ruler.
  2. Mark point ZZ anywhere above XYXY.
  3. Take any point AA on XYXY and join AZAZ.
  4. At ZZ, construct ∠BZA\angle BZA equal to ∠ZAY\angle ZAY such that BB and YY are on opposite sides of AZAZ.
  5. Extend BZBZ to form line mm. Since alternate interior angles are equal, m∥XYm \parallel XY.

Explanation:

The construction relies on the property that if a transversal (AZAZ) makes equal alternate interior angles (∠ZAY\angle ZAY and ∠BZA\angle BZA), then the lines (mm and XYXY) must be parallel.

Problem 4:

Given a line pp and a point MM at a distance of 3.5 cm3.5\text{ cm} from it, draw a line qq parallel to pp passing through MM using the perpendicular method.

Construction of parallel lines using double perpendiculars at N and M.

Solution:

  1. Draw line pp.
  2. Take any point NN on pp and draw a perpendicular line nn at NN using a compass or set-square.
  3. Measure 3.5 cm3.5\text{ cm} on this perpendicular line starting from NN and mark point MM.
  4. At point MM, draw another perpendicular to line nn.
  5. This new line is the required line qq, which is parallel to pp.

Explanation:

If two lines are both perpendicular to the same line, they are parallel to each other. Here, both pp and qq are perpendicular to the line MNMN.