The combination of two or more lines that are stretched to infinity and never intersect each other is called parallel lines or coplanar lines. The parallel lines are denoted by a special symbol, given by ||.
Properties:
- Two lines are said to be parallel if they maintain a constant perpendicular distance between them
- The slope of two parallel lines are equal
- They never intersect each other
- They always have an equal base angle to any axis.
If lines a and b are parallel, then ∠A = ∠B
Angles related to Parallel Lines
- Alternate angles: They are the angles that are formed when a line crosses two lines, that lie on opposite sides of the transversal line and opposite relative sides of the other lines. If the two lines are equal, then the alternate angles are also equal.
Here, alternate angles are equal, ∠A = ∠B
- Corresponding angles: The angles made when a transversal intersect with any pair of lines are called the corresponding angles. If the two lines are parallel, then the corresponding angles are also equal.
Here, corresponding angles are equal, ∠A = ∠B
- Interior angles: If a traversal line intersects two parallel lines, then the sum of each pair of internal angles on the same side of the traversal line is supplementary.
∠A + ∠B = 180o
Slope of the Lines
The slope is a measurement to determine the extent line is oriented to an axis. In general, we define slope as the tangent of the angle between the X-axis and a line. If the angle between X-axis and a given line is θ, then
Here, tanθ is slope and b is the intercept
Slope = tanθ
1. For a line with equation y = mx + b, we define:
- m: Slope of the given line
- b: Intercept of the line on the Y-axis.
2. If two points (a,b) and (c,d) on the line are given, then we can find the slope of the line using the formula:
(a,b) and (c,d) are two points on the line
Slope = (b – d) / (a – c)
The equation of the line can be determined by considering a point (x, y) on the line. Since we can also find the slope in terms of x and y, we can write
Slope = (y – d) / (x – c)
Equating this with the actual value of the slope, we can derive the equation for the given line as,
(y – d) / (x – c) = (b – d) / (a – c)
3. For a given line ax + by + c = 0, we can write the slope as:
Slope = – a / b
Sample Problems
Question 1. Are the lines y = 2x + 3 and 5y = 10x + 21 parallel?
Solution:
Given the lines are
y = 2x + 3 — (i)
5y = 10x + 21 — (ii)
Dividing equation (ii) with 5, we get:
5y/5 = 10x/5 + 21/5
y = 2x + 4.2
Thus, we see that the slope of both the lines are equal to 2, hence the lines are parallel.
Question 2. Determine the slope of the line 2x + 5y + 6 = 0.
Solution:
Since, the given line is in the form of ax + by + c = 0, we can compare the values of a , b and c and get their values as,
a = 2, b = 5 and c = 6
Thus, the slope of the line = -a/b = – 2/5 = -0.40
Question 3: Determine the equation of a line that passes through (1, 2) and is parallel to the line y = 2x + 5.
Solution:
The given line y = 2x + 5 has slope equals to 2.
We consider any point (x, y) on the line of which we need to find the equation. So, we can write the slope of the line as:
Slope = (y – 2)/(x – 1)
Since this line is parallel to y = 2x + 5, ao the slope of the line is also equal to 2. So, we can write:
(y – 2) / (x – 1) = 2y – 2 = 2x – 2
y = 2x
So, the equation of the line is y = 2x
Question 4. If a line passes through the points (k, 2) and (6, 7) and is parallel to the line 3x – 3y + 5 = 0, then find the value of k.
Solution:
Given line is parallel to the line 3x – 3y + 5 = 0. Since, this is in the form of ax + by + c = 0, we can write,
Slope of this line = – a / b = – 3 / (-3) = 1
In terms of the two points, we can write the slope of the given line as,
Slope = (7 – 2) / (6 – k) = 5 / (6 – k)
Since, the lines are parallel, we can write:
5 / (6 – k) = 1
6 – k = 5
k = 1
Hence, the value of k is 1.
Question 5. A line is parallel to the line y = x + 3. Find the internal angle between the line and X-axis.
Solution:
Since the lines are parallel, they must have the same slope.
So, the slope of the given line = Slope of y = x + 3 = 1
Lets, the angle made by the line and X-axis be θ, then we can write:
tanθ = 1
θ = tan-1(1) =45o
So, the angle between the line and X-axis is equal to 45o.