Chapter 10 Brain Teasers | Parallel Lines | Class-8 DAV Secondary Mathematics

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 Unit 10 Brain Teaser | Parallel Lines | Class-8 DAV Secondary Mathematics

Q1A. Tick (✓) the correct option.

(i) A transversal intersects two lines. Which condition would always make the two lines parallel?

(a) Vertically opposite angles are equal.

(b) Interior angles on the same side of the transversal are complementary.

(c) Alternate interior angles are equal.

(d) Corresponding angles are supplementary.

(ii) Based on the diagram given below, which of the following statements is true?

(a) a || b (b) a || c (c) b || c (d) d || e

(iii) In the diagram given below, transversal TU intersects PQ and RS at V and W respectively. For which value of x is PQ || RS?

(a) 6 (b) 16 (c) 24 (d) 28

(iv) Peach Steet and Cherry Street are parallel. Apple Street intersects them, as shown below:

If m∠1 = (2x + 36)° , m∠2 = (7x-9)° , what is m∠1?

(a) 9° (b) 17° (c) 54° (d) 70°

Answer 1A (iv):

In the figure, ∠2 sits above Cherry Street to the right of Apple Street. The angle in the same position at Peach Street is equal to ∠2 (corresponding angles), and it makes a straight line with ∠1. So ∠1 + ∠2 = 180°.

(2x + 36) + (7x − 9) = 180 ⇒ 9x + 27 = 180 ⇒ 9x = 153 ⇒ x = 17.

m∠1 = 2 × 17 + 36 = 70°, so the answer is (d) 70°. (54° would come from treating the angles as equal, which does not fit this figure; check: ∠2 = 7 × 17 − 9 = 110° and 70° + 110° = 180° ✓)

(v) In the diagram given below, lines n and m are cut by transversal p and q?

What value of x would make lines n and m parallel?

(a) 110° (b) 80° (c) 70° (d) 50°

Q1B. Answer the following questions.

(i) If AB || DE, find ∠ACD.

(ii) In the diagram given below, PQ || OL.

Is OL || ST? Give reasons.

(iii) If AB || PQ, find y.

(iv) if AB || EF, DE || BC, AC || Df, find x and y.

(v) For what value of x, is l || m?

Q2. In the figure 10.34, lines I II m II n and p II q II r. If ∠I = 85°, find ∠x, ∠y, ∠z, ∠r, ∠s and ∠u.

Q3. Draw a line segment AB = 7.6 cm. Without constructing parallel lines at A and B, find three points P, Q and R on AB such that AP = PQ = QR = RB.

Q4. In the figure 10.35, name the edges of the adjoining cube which are parallel to

(i) AB (ii) EH (iii) AE

(iv) What is the point of intersection of AE and AB?

(v) Are the edges AB and DH parallel to each other? Why?

(vi) Are edges EF and BC parallel?

[Lines in the same plane are either parallel or intersecting whereas lines in different planes need not be sol

HOTS (Higher Order Thinking Skills) Questions

HOTS 1: In the given figure, line l ∥ m and m ∥ n. If y : z = 3 : 7, find the measure of angle x.
Solution:
Since l ∥ m and m ∥ n, line l is parallel to line n.
Therefore, x = z (alternate interior angles).
Also, since l ∥ m, the consecutive interior angles on the same side of the transversal sum to 180°:
x + y = 180° → z + y = 180°.
Given ratio y : z = 3 : 7:
Let y = 3k and z = 7k.
3k + 7k = 180° → 10k = 180° → k = 18°.
z = 7 × 18° = 126°.
Since x = z: x = 126°.

HOTS 2: Prove that if two parallel lines are intersected by a transversal, the bisectors of any pair of alternate interior angles are parallel to each other.
Proof:
Let AB ∥ CD and transversal PQ intersect AB at E and CD at F.
Let alternate interior angles be ∠AEF and ∠EFD. Since AB ∥ CD, ∠AEF = ∠EFD.
Let EG and FH be the angle bisectors of ∠AEF and ∠EFD respectively.
Then ∠GEF = (1/2) ∠AEF, and ∠EFH = (1/2) ∠EFD.
Since ∠AEF = ∠EFD → (1/2) ∠AEF = (1/2) ∠EFD → ∠GEF = ∠EFH.
Notice that ∠GEF and ∠EFH are alternate interior angles formed by transversal EF with lines EG and FH.
Since these alternate interior angles are equal, by converse of alternate interior angles theorem:
EG ∥ FH. (Hence Proved!)

Value Based Questions (VBQs)

Question: Railway engineers must ensure that railway tracks remain strictly equidistant (parallel) throughout their course across curves, bridges, and junctions.

(a) What geometric condition guarantees that two rails never intersect?
Solution:
The perpendicular distance between the two lines must remain constant everywhere along the track.

(b) What value is demonstrated by engineers maintaining strict standards in railway track alignment?
Solution:
Passenger safety, diligence, professional precision, and accountability to public wellbeing.