Parallel and Intersecting Lines | FIO

Question 10

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

Question diagram 1
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Solution
Understand the Question
  • Angle Sum Property: The sum of angles in any triangle is always 180∘180^\circ.
  • Parallel Lines and Transversals:
    • Corresponding angles are equal when two parallel lines are cut by a transversal.
    • Alternate interior angles are equal when two parallel lines are cut by a transversal.
  • Vertically Opposite Angles: When two lines intersect, the vertically opposite angles are equal.

(i) Find the angles xx and yy in the first figure.

Step 1 · Find Angle xx

Diagram 1

The vertical line meets the parallel lines at a right angle (90∘90^\circ).

Using the angle sum property in the bottom triangle:

x∘+65∘+90∘=180∘x∘+155∘=180∘x∘=180∘−155∘x=25∘\begin{aligned} x^\circ + 65^\circ + 90^\circ &= 180^\circ \\ x^\circ + 155^\circ &= 180^\circ \\ x^\circ &= 180^\circ - 155^\circ \\ x &= 25^\circ \end{aligned}

Step 2 · Find Angle yy

Diagram 2

The angle formed at the bottom line is the sum of 90∘90^\circ and 65∘65^\circ: 90∘+65∘=155∘90^\circ + 65^\circ = 155^\circ

Since corresponding angles between parallel lines are equal: y=155∘y = 155^\circ

Answer

(i) x=25∘,  y=155∘x = 25^\circ,\; y = 155^\circ

(ii) Find the angle xx in the second figure.

Step 1 · Find Angle xx

Diagram 3

Using alternate interior angles for both transversals: α1=53∘\alpha_1 = 53^\circ α2=78∘\alpha_2 = 78^\circ

The angle between the two transversal lines is:

αx=α2−α1=78∘−53∘=25∘\begin{aligned} \alpha_x &= \alpha_2 - \alpha_1 \\ &= 78^\circ - 53^\circ \\ &= 25^\circ \end{aligned}

Since vertically opposite angles are equal: x=αx=25∘x = \alpha_x = 25^\circ

Answer

(ii) x=25∘x = 25^\circ

Common Mistakes
  • Confusing Corresponding and Alternate Angles: Always check whether the angles lie on the same relative side of the transversal (corresponding) or on opposite interior sides (alternate interior).
  • Missing the Composite Angle: In the first figure, remember that the total angle on the bottom parallel line is the sum of the right angle (90∘90^\circ) and the given angle (65∘65^\circ).

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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