Parallel and Intersecting Lines | FIO

Question 1

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

Question diagram 1Question diagram 2
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Solution
Understand the Question
  • Linear Pair: A pair of adjacent angles whose non-common arms form a straight line. Their sum is always 180∘180^\circ.
  • Vertically Opposite Angles: Pairs of non-adjacent angles formed directly opposite each other when two straight lines intersect. They are always equal in measure.

Step 1 · Identify Linear Pairs

A linear pair consists of two adjacent angles that lie on a straight line.Diagram 1

From the figure, the pairs of adjacent angles forming a straight line are:

  • ∠a\angle a and ∠b\angle b (on line mm)
  • ∠b\angle b and ∠c\angle c (on line ll)
  • ∠c\angle c and ∠d\angle d (on line mm)
  • ∠d\angle d and ∠a\angle a (on line ll)

Step 2 · Identify Vertically Opposite Angles

Vertically opposite angles are directly opposite to each other at the point of intersection of two lines.Diagram 2

From the figure, the vertically opposite pairs are:

  • ∠a\angle a and ∠c\angle c
  • ∠b\angle b and ∠d\angle d
Answer
  • Linear Pairs: ∠a\angle a and ∠b\angle b, ∠b\angle b and ∠c\angle c, ∠c\angle c and ∠d\angle d, ∠d\angle d and ∠a\angle a
  • Vertically Opposite Angles: ∠a\angle a and ∠c\angle c, ∠b\angle b and ∠d\angle d
Common Mistakes
  • Confusing Linear Pairs with Opposite Angles: Linear pairs are adjacent (side-by-side) angles on a line summing to 180∘180^\circ, whereas vertically opposite angles lie directly across the vertex from each other.
  • Missing Pairs: Forgetting that an intersection of two lines produces four linear pairs in total (each angle forms a pair with each of its two immediate neighbors).

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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