Parallel lines are straight lines that lie in the same plane and never intersect, no matter how far they are extended in either direction. The distance between parallel lines remains constant throughout.
In the figure above, the pairs of lines a and b and x and y are parallel. This is denoted by the symbol || and written as "a || b" and "x || y." Such lines do not meet at any point, even when extended indefinitely.
Theorem: Lines Parallel to the Same Line
Statement: If two lines are parallel to the same line, then they are parallel to each other.
This theorem explains the relationship between parallel lines in a plane. When two different lines are each parallel to a third line, they never intersect that line. As a result, the two given lines also do not intersect each other and hence remain parallel. This property is commonly used in geometry to establish parallelism between lines.
Proof:
From the given figure, m || l and n || l.
Line t intersects lines l, m, and n and hence acts as a transversal.
Since m || l, the corresponding angles formed by transversal t are equal.
Therefore, ∠1 = ∠2.
Similarly, since n || l, corresponding angles are equal.
Therefore, ∠1 = ∠3.
From the above results,
∠2 = ∠3.
Now, ∠2 and ∠3 are corresponding angles formed by transversal t on lines m and n.
Since corresponding angles are equal, by the converse of the corresponding angles postulate,
m || n.
Hence proved.
Note: This property can be extended to more than two lines as well.
Example 1: From the given figure below, AB ∥ || CD and CD ∥ || EF. Also, EA ⟂ AB.
If ∠BEF = 55°, find the values of x, y, and z.
Solution:
Given that AB || CD, CD || EF, EA ⟂ AB, and ∠BEF = 55°.
Since line BE intersects the parallel lines CD and EF, it acts as a transversal. Therefore, interior angles on the same side of the transversal are supplementary.
So,
y + 55° = 180°
y = 125°.Since AB || CD, corresponding angles are equal.
Therefore,
x = y = 125°.Now, AB || EF and EA ⟂ AB, so ∠EAB = 90°.
At point E, the angles formed on the straight line EF are supplementary.
55° + z + 90° = 180°
z = 35°.Therefore, the values of x, y and z are 125°, 125° and 35°, respectively.
Example 2: Consider the following figure in which L₁ || L₂.
The angle bisectors of ∠BAX and ∠ABY intersect at point C, as shown in the figure below.
Prove that ∠ACB = 90°.
Solution:
Since L₁ || L₂, the angles ∠BAX and ∠ABY are co-interior angles.
Co-interior angles between parallel lines are supplementary.So,
∠BAX + ∠ABY = 180°The lines drawn at A and B bisect these angles.
Let the two bisected angles be ∠1 and ∠2.Therefore,
2(∠1 + ∠2) = 180°∠1 + ∠2 = 90°
Now, consider triangle ACB.
By the angle sum property of a triangle,∠1 + ∠2 + ∠ACB = 180°
Substituting the value of ∠1 + ∠2,
90° + ∠ACB = 180°
∠ACB = 90°
Hence, ∠ACB is a right angle.
Practice Problems - Unsolved
Q1. Consider the following figure, in which L1 and L2 are parallel lines. What is the value of ∠C?
Q2. Consider the following figure, in which AX ∥ DY. What is the difference between ∠1 and ∠2?
Q3. From the given figure, if PQRS and ∠MXQ = 120° and ∠MYR = 50°, then find the value of ∠XMY.
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