Page 117 - classs 6 a_Neat
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x
( )
∠A =
°
2
EXAMPLE 5: In a ∆ABC, if 2 A= 3 B = 6 C then calculate A, B and C.
Let, 2 A = 3 B = 6 C = x0
° ° °
x
x
x
∠ A = ; ∠ B = and C = ,
∠
3
2
6
∠
180 .
but , A +∠ C = °
B +∠
x
x
x
=
+
+
∴ 180
2 3 6
x +
⇒ 3 2 1080 ⇒ 6 1080 ⇒ = x 1080 180.
=
x =
x =
x +
6
° 180 ° x °
180
°
∠ A = = 90 ; ∠ ° B = = 60 ° and ∠ C = = 30 ,
2 3 6 PARALLEL LINES Two lines in a plane which do not meet even when produced indefinitely in either
°
°
∠
∠
∠
Hence , A = 90 B = 60 C = 30 ° .
direction, are known as parallel lines.
EXAMPLE 6: The adjoining figure has been obtained by using two triangles
If l and m are two parallel lines, we write l || m and read it
as l is parallel tom.
Prove that A + B + C + D + E + F = 360° . Clearly, when l || m, we have, m || l
We know that the sum of the angles of a triangle is 180° .
In ∆ACE, we have:
A + C + E = 180° . DISTANCE BETWEEN TWO PARALLEL LINES
In ∆BDF, we have: Let us draw two lines l and m such that l || m.
Take any point A on one of these lines, say l. At A,
B + D + F = 180° . draw AD perpendicular to l, meeting m at D.
Adding the corresponding sides of the above equations, we get: Measure the length of the line segment AD.
A + B + C + D + E + F = 360° . This length AD is called the perpendicular distance between l and m at the point A.
PARALLEL LINES Let us take any other point B on l. From B, draw BC
perpendicular to l, meeting m at C. Measure the length of the line
Let us draw two straight lines AB and CD, as shown in the figure segment BC.
(i). We find that these lines when produced towards the left, meet at a point 0.
We find that AD = BC.
Thus, the perpendicular distance between l and m at the point A is
the same as that at B.
Actually speaking, the perpendicular distance between two parallel lines
is the same throughout. This distance is called the distance between two parallel lines.
Again, let us draw straight lines PQ and RS, as shown in the figure
Thus, parallel lines are the same distance apart throughout.
(ii). These lines, when produced towards the right, meet at a point M.
Why railway lines are made parallel The wheels of a railway engine and those of the bogies are attached by
However, there are examples of lines which when produced indefinitely in either direction, do not meet. Such axles of a fixed length.
lines are known as parallel lines. The opposite edges of a blackboard, the opposite edges of a ruler, railway
lines, etc., are all examples of parallel lines. So, the distance between each pair of opposite wheels remains fixed.
Therefore, the rails on which these wheels roll, must be at a constant distance from each other.
Hence, the opposite rails must be parallel.
