Parallel and Intersecting Lines | FIO

Question 4

In the dot paper following, draw different sets of parallel lines. The line segments can be of different lengths but should have dots as endpoints.

Question diagram 1
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Solution
Understand the Question
  • Parallel lines are lines in a plane that never meet or intersect, maintaining a constant distance between them everywhere.
  • On dot paper (grid paper), lines are parallel if they have the same steepness or slope (the ratio of vertical change rise\text{rise} to horizontal change run\text{run}): Slope=RiseRun\text{Slope} = \dfrac{\text{Rise}}{\text{Run}}
  • We can draw different sets of parallel lines by choosing:
    • Horizontal lines: Same vertical position across columns (slope =0= 0).
    • Vertical lines: Same horizontal position across rows (slope is undefined).
    • Diagonal lines: Identical rise-over-run ratios between endpoints.

Step 1 · Draw Horizontal Parallel Lines

Horizontal lines connect dots along the same horizontal row, so their vertical change (rise) is 00.Diagram 1

  • Line 1: Connect dot (1,6)(1, 6) to dot (5,6)(5, 6)
  • Line 2: Connect dot (1,4)(1, 4) to dot (5,4)(5, 4)

Both lines are horizontal with slope =0= 0, so they are parallel.

Step 2 · Draw Vertical Parallel Lines

Vertical lines connect dots along the same vertical column, having a horizontal change (run) of 00.Diagram 2

  • Line 1: Connect dot (2,8)(2, 8) to dot (2,5)(2, 5)
  • Line 2: Connect dot (4,8)(4, 8) to dot (4,5)(4, 5)

Both lines are vertical and maintain a constant distance of 22 units, so they are parallel.

Step 3 · Draw Diagonal Parallel Lines

Diagonal lines are parallel if their slope (RiseRun)\left(\dfrac{\text{Rise}}{\text{Run}}\right) is the same.Diagram 3

For a slope of 11 (move 11 unit up for every 11 unit right):

  • Line 1: Connect dot (1,5)(1, 5) to dot (4,8)(4, 8)
  • Line 2: Connect dot (3,3)(3, 3) to dot (6,6)(6, 6)

Slope=8−54−1=33=1,Slope=6−36−3=33=1\text{Slope} = \dfrac{8 - 5}{4 - 1} = \dfrac{3}{3} = 1, \quad \text{Slope} = \dfrac{6 - 3}{6 - 3} = \dfrac{3}{3} = 1

Since both lines have equal slopes, they are parallel.

Answer

Different sets of parallel lines can be formed on dot paper by:

  1. Connecting dots horizontally in different rows.
  2. Connecting dots vertically in different columns.
  3. Connecting dots diagonally with the same ratio of RiseRun\dfrac{\text{Rise}}{\text{Run}}.
Common Mistakes
  • Unequal Slopes for Diagonals: Drawing slanted lines by eyeballing rather than counting dots (e.g., pairing a line going 11 up, 22 right with one going 11 up, 33 right), resulting in lines that will eventually intersect.
  • Off-Grid Endpoints: Ending line segments in the space between dots instead of on exact dot coordinates as specified by the question.

More questions in FIO

Q1

List all the linear pairs and vertically opposite angles you observe in Fig. 5.3:

Q2

Draw some lines perpendicular to the lines given on the dot paper in Fig. 5.10.

Q3

In Fig. 5.11, mark the parallel lines using the notation given above (single arrow, double arrow etc.). Mark the angle between perpendicular lines with a square symbol.

(a) How did you spot the perpendicular lines?

(b) How did you spot the parallel lines?

Q4

In the dot paper following, draw different sets of parallel lines. The line segments can be of different lengths but should have dots as endpoints.

Q5

Using your sense of how parallel lines look, try to draw lines parallel to the line segments on this dot paper.

(a) Did you find it challenging to draw some of them? (b) Which ones? (c) How did you do it?

Q6

In Fig. 5.13, which line is parallel to line aa — line bb or line cc? How do you decide this?

Q7

Can you draw a line parallel to ll, that goes through point AA? How will you do it with the tools from your geometry box? Describe your method.

Q8

Find the angles marked below.

Q9

Find the angle represented by aa.

Q10

In the figures below, what angles do xx and yy stand for?

Q11

In Fig. 5.33, ∠ABC=45∘\angle ABC = 45^\circ and ∠IKJ=78∘\angle IKJ = 78^\circ. Find angles ∠GEH\angle GEH, ∠HEF\angle HEF, ∠FED\angle FED

Q12

In Fig. 5.34, ABAB is parallel to CDCD and CDCD is parallel to EFEF. Also, EAEA is perpendicular to ABAB. If ∠BEF=55∘\angle BEF = 55^\circ, find the values of xx and yy.

Q13

What is the measure of angle ∠NOP\angle \text{NOP} in Fig. 5.35?

[Hint: Draw lines parallel to LM and PQ through points N and O.]

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