Parallel and Intersecting Lines | FIO

Question 8

Find the angles marked below.

Question diagram 1
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Solution
Understand the Question

To find the unknown angles, we apply the geometric relationships between angles formed by parallel lines cut by transversals, as well as fundamental angle rules:

  • Alternate interior angles are equal.
  • Corresponding angles are equal.
  • Consecutive interior angles (co-interior) add up to 180∘180^\circ.
  • Vertically opposite angles are equal.
  • Linear pair angles on a straight line sum to 180∘180^\circ.
  • Angle sum property of a triangle: The sum of all interior angles in any triangle is 180∘180^\circ.

(a) Find angle aa

Step 1 · Find Angle aa

Diagram 1Angle aa and the 48∘48^\circ angle are alternate interior angles. a=48∘a = 48^\circ

Answer

(a) 48∘48^\circ

(b) Find angle bb

Step 1 · Find Angle bb

Diagram 2Angle bb and the 52∘52^\circ angle are alternate interior angles. b=52∘b = 52^\circ

Answer

(b) 52∘52^\circ

(c) Find angle cc

Step 1 · Find Angle cc

Diagram 3Let the angle adjacent to 99∘99^\circ on the straight line be xx.

x=180∘−99∘=81∘\begin{aligned} x &= 180^\circ - 99^\circ \\[0.6em] &= 81^\circ \end{aligned}

Angle xx and angle cc are corresponding angles. c=81∘c = 81^\circ

Answer

(c) 81∘81^\circ

(d) Find angle dd

Step 1 · Find Angle dd

Diagram 4Angle dd and the 81∘81^\circ angle are consecutive interior angles, which sum to 180∘180^\circ.

d+81∘=180∘d=180∘−81∘=99∘\begin{aligned} d + 81^\circ &= 180^\circ \\[0.6em] d &= 180^\circ - 81^\circ \\[0.6em] &= 99^\circ \end{aligned}
Answer

(d) 99∘99^\circ

(e) Find angle ee

Step 1 · Find Angle ee

Diagram 5Angle ee and the 69∘69^\circ angle are alternate interior angles. e=69∘e = 69^\circ

Answer

(e) 69∘69^\circ

(f) Find angle ff

Step 1 · Find Angle ff

Diagram 6Angle ff and the 132∘132^\circ angle are consecutive interior angles, which sum to 180∘180^\circ.

f+132∘=180∘f=180∘−132∘=48∘\begin{aligned} f + 132^\circ &= 180^\circ \\[0.6em] f &= 180^\circ - 132^\circ \\[0.6em] &= 48^\circ \end{aligned}
Answer

(f) 48∘48^\circ

(g) Find angle gg

Step 1 · Find Angle gg

Diagram 7Angle gg and the 122∘122^\circ angle are vertically opposite angles. g=122∘g = 122^\circ

Answer

(g) 122∘122^\circ

(h) Find angle hh

Step 1 · Find Angle hh

Diagram 8Find the interior angle adjacent to 120∘120^\circ on a straight line: 180∘−120∘=60∘180^\circ - 120^\circ = 60^\circ

Find the interior angle adjacent to 75∘75^\circ on a straight line: 180∘−75∘=105∘180^\circ - 75^\circ = 105^\circ

Using the angle sum property of the triangle (180∘180^\circ):

h+60∘+105∘=180∘h+165∘=180∘h=180∘−165∘=15∘\begin{aligned} h + 60^\circ + 105^\circ &= 180^\circ \\[0.6em] h + 165^\circ &= 180^\circ \\[0.6em] h &= 180^\circ - 165^\circ \\[0.6em] &= 15^\circ \end{aligned}
Answer

(h) 15∘15^\circ

(i) Find angle ii

Step 1 · Find Angle ii

Using the angle sum property of the triangle:

i+70∘+56∘=180∘i+126∘=180∘i=180∘−126∘=54∘\begin{aligned} i + 70^\circ + 56^\circ &= 180^\circ \\[0.6em] i + 126^\circ &= 180^\circ \\[0.6em] i &= 180^\circ - 126^\circ \\[0.6em] &= 54^\circ \end{aligned}
Answer

(i) 54∘54^\circ

(j) Find angle jj

Step 1 · Find Angle jj

Angle jj and the 97∘97^\circ angle are alternate interior angles. j=97∘j = 97^\circ

Answer

(j) 97∘97^\circ

Common Mistakes
  • Alternate vs. Consecutive Interior Angles: Alternate interior angles (ZZ-pattern) are equal, whereas consecutive interior angles (CC-pattern) are supplementary (sum to 180∘180^\circ).
  • Linear Pair Assumption: Assuming adjacent angles are equal instead of summing them to 180∘180^\circ when on a straight line.

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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