Parallel and Intersecting Lines | IT

Question 12

What about five different angles — 6, 5, 4, 3 and 2?

Question diagram 1
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Solution
Understand the Question
  • When a transversal intersects two lines ll and mm, it creates eight angles (11 to 88).
  • General angle properties (always true):
    • Linear Pair: Adjacent angles on a straight line add up to 180∘180^\circ.
    • Vertically Opposite Angles: Non-adjacent angles formed by two intersecting lines are equal.
  • Parallel line properties (hold only when l∥ml \parallel m):
    • Corresponding Angles: In the same relative position at each intersection, and are equal.
    • Alternate Angles (Interior/Exterior): On opposite sides of the transversal, and are equal.
    • Consecutive Interior Angles: On the same side of the transversal inside the two lines, and sum to 180∘180^\circ.

(i) Angle 6

Step 1 · Analyze Angle 6

Diagram 1

  • ∠6\angle 6 is an interior angle.
  • Linear pair: ∠6+∠5=180∘\angle 6 + \angle 5 = 180^\circ
  • Vertically opposite angle: ∠6=∠8\angle 6 = \angle 8

If lines ll and mm are parallel (l∥ml \parallel m):

  • Corresponding angle: ∠6=∠2\angle 6 = \angle 2
  • Alternate interior angle: ∠6=∠3\angle 6 = \angle 3
  • Consecutive interior angles: ∠6+∠4=180∘\angle 6 + \angle 4 = 180^\circ
Answer

(i) ∠6\angle 6 is an interior angle. ∠6+∠5=180∘\angle 6 + \angle 5 = 180^\circ (linear pair), ∠6=∠8\angle 6 = \angle 8 (vertically opposite). If l∥ml \parallel m: ∠6=∠2\angle 6 = \angle 2 (corresponding), ∠6=∠3\angle 6 = \angle 3 (alternate interior), and ∠6+∠4=180∘\angle 6 + \angle 4 = 180^\circ (consecutive interior).

(ii) Angle 5

Step 1 · Analyze Angle 5

  • ∠5\angle 5 is an interior angle.
  • Linear pair: ∠5+∠6=180∘\angle 5 + \angle 6 = 180^\circ
  • Vertically opposite angle: ∠5=∠7\angle 5 = \angle 7

If lines ll and mm are parallel (l∥ml \parallel m):

  • Corresponding angle: ∠5=∠4\angle 5 = \angle 4
  • Alternate interior angle: ∠5=∠1\angle 5 = \angle 1
  • Consecutive interior angles: ∠5+∠3=180∘\angle 5 + \angle 3 = 180^\circ
Answer

(ii) ∠5\angle 5 is an interior angle. ∠5+∠6=180∘\angle 5 + \angle 6 = 180^\circ (linear pair), ∠5=∠7\angle 5 = \angle 7 (vertically opposite). If l∥ml \parallel m: ∠5=∠4\angle 5 = \angle 4 (corresponding), ∠5=∠1\angle 5 = \angle 1 (alternate interior), and ∠5+∠3=180∘\angle 5 + \angle 3 = 180^\circ (consecutive interior).

(iii) Angle 4

Step 1 · Analyze Angle 4

  • ∠4\angle 4 is an interior angle.
  • Linear pair: ∠4+∠1=180∘\angle 4 + \angle 1 = 180^\circ
  • Vertically opposite angle: ∠4=∠2\angle 4 = \angle 2

If lines ll and mm are parallel (l∥ml \parallel m):

  • Corresponding angle: ∠4=∠5\angle 4 = \angle 5
  • Alternate interior angle: ∠4=∠8\angle 4 = \angle 8
  • Consecutive interior angles: ∠4+∠6=180∘\angle 4 + \angle 6 = 180^\circ
Answer

(iii) ∠4\angle 4 is an interior angle. ∠4+∠1=180∘\angle 4 + \angle 1 = 180^\circ (linear pair), ∠4=∠2\angle 4 = \angle 2 (vertically opposite). If l∥ml \parallel m: ∠4=∠5\angle 4 = \angle 5 (corresponding), ∠4=∠8\angle 4 = \angle 8 (alternate interior), and ∠4+∠6=180∘\angle 4 + \angle 6 = 180^\circ (consecutive interior).

(iv) Angle 3

Step 1 · Analyze Angle 3

  • ∠3\angle 3 is an interior angle.
  • Linear pair: ∠3+∠2=180∘\angle 3 + \angle 2 = 180^\circ
  • Vertically opposite angle: ∠3=∠1\angle 3 = \angle 1

If lines ll and mm are parallel (l∥ml \parallel m):

  • Corresponding angle: ∠3=∠7\angle 3 = \angle 7
  • Alternate interior angle: ∠3=∠6\angle 3 = \angle 6
  • Consecutive interior angles: ∠3+∠5=180∘\angle 3 + \angle 5 = 180^\circ
Answer

(iv) ∠3\angle 3 is an interior angle. ∠3+∠2=180∘\angle 3 + \angle 2 = 180^\circ (linear pair), ∠3=∠1\angle 3 = \angle 1 (vertically opposite). If l∥ml \parallel m: ∠3=∠7\angle 3 = \angle 7 (corresponding), ∠3=∠6\angle 3 = \angle 6 (alternate interior), and ∠3+∠5=180∘\angle 3 + \angle 5 = 180^\circ (consecutive interior).

(v) Angle 2

Step 1 · Analyze Angle 2

  • ∠2\angle 2 is an exterior angle.
  • Linear pair: ∠2+∠1=180∘\angle 2 + \angle 1 = 180^\circ
  • Vertically opposite angle: ∠2=∠4\angle 2 = \angle 4

If lines ll and mm are parallel (l∥ml \parallel m):

  • Corresponding angle: ∠2=∠6\angle 2 = \angle 6
  • Alternate exterior angle: ∠2=∠7\angle 2 = \angle 7
  • Supplementary angle: ∠2+∠5=180∘\angle 2 + \angle 5 = 180^\circ
Answer

(v) ∠2\angle 2 is an exterior angle. ∠2+∠1=180∘\angle 2 + \angle 1 = 180^\circ (linear pair), ∠2=∠4\angle 2 = \angle 4 (vertically opposite). If l∥ml \parallel m: ∠2=∠6\angle 2 = \angle 6 (corresponding), ∠2=∠7\angle 2 = \angle 7 (alternate exterior), and ∠2+∠5=180∘\angle 2 + \angle 5 = 180^\circ.

Common Mistakes
  • Assuming Parallel Lines: Linear pairs and vertically opposite angles are always valid for any intersecting lines, but corresponding, alternate, and consecutive interior relationships only hold when the lines are parallel (l∥ml \parallel m).
  • Equal vs. Supplementary: Alternate interior angles are equal (e.g., ∠6=∠3\angle 6 = \angle 3), whereas consecutive interior angles are supplementary (e.g., ∠6+∠4=180∘\angle 6 + \angle 4 = 180^\circ).

More questions in IT

Q1

Context: Let us observe what happens when two lines intersect.

Q. How many angles do they form?

Q2

Can two straight lines intersect at more than one point?

Q3

In Fig. 5.2, if ∠a\angle a is 120∘120^\circ, can you figure out the measurements of ∠b\angle b, ∠c\angle c and ∠d\angle d, without drawing and measuring them?

Q4

Context: When two lines intersect each other and form four angles, labelled aa, bb, cc and dd, then ∠a\angle a and ∠c\angle c are equal, and ∠b\angle b and ∠d\angle d are equal.

Q. Is this always true for any pair of intersecting lines?

Q5

Can you draw a pair of intersecting lines such that all four angles are equal? Can you figure out what will be the measure of each angle?

Q6

Observe Fig. 5.5 and describe the way the line segments meet or cross each other in each case, with appropriate mathematical words (a point, an endpoint, the midpoint, meet, intersect) and the degree measure of each angle.

Q7

Are line segments ST\text{ST} and UV\text{UV} likely to meet if they are extended?

Q8

Are line segments OP\text{OP} and QR\text{QR} likely to meet if they are extended?

Q9

Name some parallel lines you can spot in your classroom.

Q10

Which pairs of lines appear to be parallel in Fig. 5.6 below?

Q11

Is it possible for all the eight angles to have different measurements? Why, why not?

Q12

What about five different angles — 6, 5, 4, 3 and 2?

Q13

Suppose, we have a transversal intersecting two parallel lines. What can be said about the corresponding angles?

Q14

Context: Activity 5 In Fig. 5.20, draw a transversal tt to the lines ll and mm such that one pair of corresponding angles is equal. You can measure the angles with a protractor.

Q. Are you finding it hard to draw a transversal such that the corresponding angles are equal?

Q15

Draw two more parallel lines using the long side of the set square as shown in Fig. 5.22. How do you know these two lines are parallel? Can you check if the corresponding angles are equal?

Q17

Why are lines ll and mm parallel to each other?

Q18

Is there a relation between ∠3\angle 3 and ∠6\angle 6? You could try to find the relationship by taking different values for ∠3\angle 3 and see what ∠6\angle 6 is. Once you find a relation, try to justify it or prove that this relation holds always.

Q19

There do not seem to be any parallel lines here. Or, are there?

What causes these illusions?

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