Page 170 - ICSE Math 7

Relation between the angles made by a transversal with parallel lines

We know that two lines which lie in the same plane and do not meet even when extended indefinitely
are known as parallel lines. u
When two parallel lines are intersected by a transversal, then 2 1 v
(a) the alternate angles thus formed are equal 3 4
(b) the corresponding angles are equal

(c) the interior angles on the same side of the transversal are supplementary 6 5 w
In the adjoining figure, lines v and w are parallel and are intersected by 7 8
the transversal u. So,
∠4 = ∠6, ∠3 = ∠5, ∠2 = ∠8, ∠1 = ∠7 (Alternate angles)
∠1 = ∠5, ∠4 = ∠8, ∠2 = ∠6, ∠3 = ∠7 (Corresponding angles)
∠4 + ∠5 = 180°, ∠3 + ∠6 = 180° (Interior angles on the same side of the transversal)

Conditions for parallelism
If two straight lines are cut by a transversal, then the lines are parallel if any of the following conditions
is satisfied:
(a) a pair of alternate angles are equal
(b) a pair of corresponding angles are equal
(c) the interior angles on the same side of the transversal are supplementary

Example 3: In the adjoining figure, line PQ is parallel to line RS and
AB is a transversal. Find angles x, y and z. A
Solution: ∠ACQ = x (Vertically opposite angles) C 100°
⇒ x = 100° P x Q
As, x = y (Alternate angles)
⇒ y = 100° y
Also, y + z = 180° (Linear pair) R D z S
⇒ 100° + z = 180° B
⇒ z = 180° – 100° = 80°

Example 4: Find the obtuse angle between the minute and hour hands of a clock at 5 o’clock. Also,
find the reflex angle between the minute and hour hands at 4 o’clock.
Solution: A complete rotation of a hand of a clock generates 360°.

\ The angle between the two hands of a clock when they point 12 1
360° 11 2
to consecutive integers = (\ There are 12 divisions) 10 obtuse
12 9 angle 3
= 30° 8 7 6 5 4

\ Obtuse angle between the minute and hour hands at 5 o’clock =
30° × 5 = 150° 11 12 1
Now, obtuse angle between the minute and hour hands at 4 o’clock = 9 10 2 3
30° × 4 = 120° 8 reflex 4
angle
\ Reflex angle between the minute and hour hands at 4 o’clock = 7 6 5
360° – 120° = 240°

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