Parallel and Intersecting Lines | FIO

Question 9

Find the angle represented by aa.

Question diagram 1
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Solution
Understand the Question

To find the unknown angle aa in each figure, we apply standard geometric properties of parallel lines intersected by transversals:

  • Corresponding angles are equal.
  • Alternate interior angles are equal.
  • Vertically opposite angles are equal.
  • Angles on a straight line (linear pair) sum to 180∘180^\circ.

(i) Find the angle represented by aa in the first diagram.

Step 1 · Calculate Angle aa

Let the top parallel line be L1L_1 and the middle parallel line be L2L_2 cut by transversal TT.Diagram 1

Since L1∥L2L_1 \parallel L_2, the corresponding angle on L2L_2 is 42∘42^\circ.

Angle a∘a^\circ and this 42∘42^\circ angle form a linear pair on L2L_2

a∘+42∘=180∘a∘=180∘−42∘a=138∘\begin{aligned} a^\circ + 42^\circ &= 180^\circ \\[0.6em] a^\circ &= 180^\circ - 42^\circ \\[0.6em] a &= 138^\circ \end{aligned}
Answer

(i) 138∘138^\circ

(ii) Find the angle represented by aa in the second diagram.

Step 1 · Calculate Angle aa

Let horizontal parallel lines be H1,H2H_1, H_2 and vertical parallel lines be V1,V2V_1, V_2.Diagram 2

Angles on a straight line add up to 180∘180^\circ

∠X=180∘−62∘=118∘\begin{aligned} \angle X &= 180^\circ - 62^\circ \\[0.6em] &= 118^\circ \end{aligned}

Since H1∥H2H_1 \parallel H_2 with transversal V2V_2, the corresponding angle ∠Y=∠X=118∘\angle Y = \angle X = 118^\circ.

Since V1∥V2V_1 \parallel V_2 with transversal H2H_2, alternate interior angles are equal

a∘=∠Y=118∘\begin{aligned} a^\circ &= \angle Y \\[0.6em] &= 118^\circ \end{aligned}
Answer

(ii) 118∘118^\circ

(iii) Find the angle represented by aa in the third diagram.

Step 1 · Calculate Angle aa

Let the parallel horizontal lines from top to bottom be L1,L2,L3,L4L_1, L_2, L_3, L_4 cut by transversal TT.Diagram 3

Vertically opposite angle at L1L_1 is ∠X=110∘\angle X = 110^\circ.

Since L1∥L2L_1 \parallel L_2, corresponding angle ∠Y=∠X=110∘\angle Y = \angle X = 110^\circ.

Subtracting the given 35∘35^\circ angle gives ∠Z\angle Z

∠Z=∠Y−35∘=110∘−35∘=75∘\begin{aligned} \angle Z &= \angle Y - 35^\circ \\[0.6em] &= 110^\circ - 35^\circ \\[0.6em] &= 75^\circ \end{aligned}

Since L3∥L4L_3 \parallel L_4, corresponding angle ∠W=∠Z=75∘\angle W = \angle Z = 75^\circ.

Angle a∘a^\circ and ∠W\angle W form a linear pair on L4L_4

a∘+∠W=180∘a∘+75∘=180∘a∘=180∘−75∘a=105∘\begin{aligned} a^\circ + \angle W &= 180^\circ \\[0.6em] a^\circ + 75^\circ &= 180^\circ \\[0.6em] a^\circ &= 180^\circ - 75^\circ \\[0.6em] a &= 105^\circ \end{aligned}
Answer

(iii) 105∘105^\circ

(iv) Find the angle represented by aa in the fourth diagram.

Step 1 · Calculate Angle aa

Let the slanted parallel lines be L1L_1 and L2L_2, intersected by horizontal line HH and vertical line VV (V⊥HV \perp H, so angle is 90∘90^\circ).Diagram 4

Angles on the straight line HH sum to 180∘180^\circ

67∘+∠X+90∘=180∘∠X+157∘=180∘∠X=180∘−157∘=23∘\begin{aligned} 67^\circ + \angle X + 90^\circ &= 180^\circ \\[0.6em] \angle X + 157^\circ &= 180^\circ \\[0.6em] \angle X &= 180^\circ - 157^\circ \\[0.6em] &= 23^\circ \end{aligned}

Since L1∥L2L_1 \parallel L_2 with transversal VV, alternate interior angles are equal

a∘=∠X=23∘\begin{aligned} a^\circ &= \angle X \\[0.6em] &= 23^\circ \end{aligned}
Answer

(iv) 23∘23^\circ

Common Mistakes
  • Confusing Angle Pairs: Misidentifying consecutive interior angles (which add up to 180∘180^\circ) as alternate interior or corresponding angles (which are equal).
  • Straight Line Sum: Forgetting that all adjacent angles on a straight line must sum to 180∘180^\circ, especially when more than two angles meet at a vertex.

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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