Parallel and Intersecting Lines | FIO

Question 13

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.]

Question diagram 1
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Solution
Understand the Question
  • Lines LM\text{LM} and PQ\text{PQ} are parallel (indicated by arrows).
  • To find the unknown angle ∠NOP\angle \text{NOP}, we draw auxiliary lines RS\text{RS} through point N\text{N} and TU\text{TU} through point O\text{O} parallel to LM\text{LM} and PQ\text{PQ}.
  • This divides the given angles into pairs of alternate interior angles, which are equal between parallel lines.

Step 1 · Draw Auxiliary Lines and Find Angles at Point N

Draw line RS\text{RS} through point N\text{N} parallel to LM\text{LM}, and line TU\text{TU} through point O\text{O} parallel to PQ\text{PQ}.

Since LM∥PQ\text{LM} \parallel \text{PQ}, all four lines are parallel: LM∥RS∥TU∥PQ\text{LM} \parallel \text{RS} \parallel \text{TU} \parallel \text{PQ}Diagram 1

Since LM∥RS\text{LM} \parallel \text{RS} with transversal MN\text{MN}, alternate interior angles are equal: ∠MNS=∠LMN=40∘  ⟹  w∘=40∘\angle \text{MNS} = \angle \text{LMN} = 40^\circ \implies w^\circ = 40^\circ

Since ∠MNO=96∘\angle \text{MNO} = 96^\circ:

w∘+x∘=96∘40∘+x∘=96∘x∘=96∘−40∘=56∘\begin{aligned} w^\circ + x^\circ &= 96^\circ \\[0.6em] 40^\circ + x^\circ &= 96^\circ \\[0.6em] x^\circ &= 96^\circ - 40^\circ = 56^\circ \end{aligned}

∠SNO=x∘=56∘\angle \text{SNO} = x^\circ = 56^\circ

Step 2 · Calculate Angle NOP

Since RS∥TU\text{RS} \parallel \text{TU} with transversal NO\text{NO}, alternate interior angles are equal: ∠NOT=∠SNO  ⟹  y∘=x∘=56∘\angle \text{NOT} = \angle \text{SNO} \implies y^\circ = x^\circ = 56^\circ

Since TU∥PQ\text{TU} \parallel \text{PQ} with transversal OP\text{OP}, alternate interior angles are equal: ∠TOP=∠OPQ  ⟹  z∘=52∘\angle \text{TOP} = \angle \text{OPQ} \implies z^\circ = 52^\circ

Now, add both parts to find ∠NOP\angle \text{NOP} (a∘a^\circ):

a∘=y∘+z∘=56∘+52∘=108∘\begin{aligned} a^\circ &= y^\circ + z^\circ \\[0.6em] &= 56^\circ + 52^\circ \\[0.6em] &= 108^\circ \end{aligned}
Answer

108∘108^\circ

Common Mistakes
  • Forgetting Auxiliary Lines: Attempting to directly add or subtract ∠LMN\angle \text{LMN}, ∠MNO\angle \text{MNO}, and ∠OPQ\angle \text{OPQ} without constructing parallel lines through vertices N\text{N} and O\text{O}.
  • Confusing Interior Angles: Mistaking alternate interior angles (which are equal) with co-interior/consecutive interior angles (which sum to 180∘180^\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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