Parallel Lines and its Properties
If the distance between two lines is always the same, the lines are said as parallel lines. These lines never meet each other. They never cross each other’s path.
We come across parallel lines in daily life, we see parallel lines in railway tracks, zebra crossing, stairs, walls, window linings, lines in notebook pages, etc.
Two lines are parallel, if they are drawn in the same plane. Let us study parallel lines and its properties.
Condition for Parallel Lines
When the two straight lines on the same plane do not intersect, they are called parallel lines.
The perpendicular distance between the two parallel lines always remains the same.
The symbol for parallel lines is ||and it is called ‘parallel to’.
Parallel lines are indicated with a pair of vertical pipes between the line names like line m is parallel to line n written as m || n
An easy way to remember parallel lines is to assume “equal to (=)” sign. Some of the condition for parallel lines are discussed below
Properties of Parallel Lines
Some of the important properties of parallel lines are:
The perpendicular distance between two parallel lines is equal everywhere.
If two lines are parallel to the same line, then all the three lines are parallel to each other.
Two lines in the same plane and perpendicular to the same line are parallel to each other.
One and only one parallel line can be drawn parallel to a given line through a given point which is not lying on the given line.
The lines that are not parallel to each other and if they cross each other they are called intersecting lines. A line that intersects two or more lines at different points is called a transversal.
A line that intersects two or more lines at different points, is called a transversal of the given lines. In the figure below the red line indicates the transversal.
The transversal intersects two lines in two points and makes 4 angles at each intersecting point. In all there are eight angles formed.
∠1, ∠2, ∠7, and ∠8 are the exterior angles because they are outside the lines.
∠3, ∠4, ∠5, and ∠6 are the interior angles because they are inside the lines.
When a transversal intersects two lines it forms some pair of angles they are as follows
Two angles on the same side of a transversal are known as the corresponding angles if both lie above the two lines or below the two lines. Pairs of corresponding angles are:
∠1 and ∠6
∠4 and ∠7
∠2 and ∠5
∠3 and ∠8
Alternate Interior Angles:
Two pairs of interior angles on the same side of the transversal are called pairs of consecutive interior angles. Pairs of interior alternate angles are:
∠4 and ∠5
∠3 and ∠6
Alternate Exterior Angles:
∠1 and ∠8
∠2 and ∠7
Same Side Interior Angles:
∠3 and ∠5
∠4 and ∠6
Now, let us study the relations between the angles of the pairs when the two lines are parallel to each other and under the condition for parallel lines.
Parallel Lines Axioms and Theorem
Consider line a||b and l is the transversal.
The following are the axioms and theorems for the parallel lines.
Corresponding Angle Axiom
If two parallel lines are intersected by a transversal, then each pair of corresponding angles are equal.
i.e. ∠1=∠6, ∠4=∠8, ∠2= ∠5 and ∠3= ∠7
Conversely, if pairs of corresponding angles are equal, then the given lines are parallel to each other.
The below theorem can be proved if we assume that one of them always holds good.
By considering the corresponding angle axiom we can prove the below theorems.
If two parallel lines are intersected by a transversal then the pair of alternate interior angles are equal.
I.e. ∠4=∠5 and ∠3=∠6
Proof: From figure 1
∠1=∠3…………..(Vertically Opposite Angles)
∠1=∠6 …………..(Corresponding Angles Axiom)
So, we get
∠4 =∠5 and ∠3 =∠6
I.e If two lines which are parallel are intersected by a transversal then the pair of alternate interior angles are equal.
Conversely, if the pair of alternate interior angles are equal then the given lines are parallel to each other.
If two parallel lines are intersected by a transversal then the pair of interior angles on the same side of the transversal are supplementary.
∠3 + ∠5 = 1800 and ∠4 = ∠6 = 1800
Proof: We have
∠4 =∠5 and
∠3 =∠6……….(Alternate interior angles)
∠3+ ∠4=180° and
∠5+∠6=180° (Linear pair axiom)
⇒∠3 + ∠5=180° and ∠4 + ∠6=180°
Conversely, if the pair of co-interior angles are supplementary then the given lines are parallel to each other.
These state the parallel lines and its properties.
Example 1: AB and CD are parallel lines. EH is a transversal.What is the measure of angle AFG?
Solution: AB CD EH is the transversal.
∠FGD + ∠DGH = 1800…….(angles are in linear pair)
∠FGD = 180 – 119
∠FGD = 610
Now ∠FGD = ∠GFA ……(pair of alternate interior angles)
Therefore, ∠AFG = 610….(because ∠FGD = 610)
Example 2: PQ and RS are parallel lines and TW is a transversal.The size of angle TUQ is (x + 12)° and the size of angle SVW is (3x + 48)°What is the value of x?
Solution: PQ RS and TW is the transversal
∠QUT = ∠UVS …..( Corresponding angles axiom)
We have ∠QUT = ( x + 12)0
Therefore ∠UVS = ( x + 12)0
Now, ∠UVS + ∠SVW = 1800…( angles forming linear pair by linear pair axiom)
We have, ∠SVW = (3x + 48)0
So, ( x + 12) + ( 3x + 48) = 180
4x + 60 = 180
4x = 180 – 60
4x = 120
x = 120/4
X = 300
AB and CD are parallel lines. EH is a transversal.The measure of angle EFB is(2x – 100)° and the measure of angle CGF is (x + 52)°What is the measure of the Angle EFB ?
AB and CD are parallel lines. EH is a transversal. What is the measure of angle DGH?