Properties of parallel and perpendicular lines worksheet is much useful to kids who would like to practice problems in Coordinate geometry.

To know some basic stuff about parallel lines,

To know some basic stuff about perpendicular lines,

1) The slopes of the two lines are 7 and (3k +2). If the two lines are parallel, find the value of "k"

2) The equations of the two parallel lines are

3x + 2y - 8 = 0

(5k+3) + 2y + 1 = 0

Find the value of "k"

3) Find the equation of a straight line is passing through (2, 3) and parallel to the line 2x - y + 7 = 0.

4) Verify, whether the two lines 3x + 2y - 7 = 0 and y = -1.5x + 4 are parallel.

5) Verify, whether the two lines 5x + 7y - 1 = 0 and 10x + 14y + 5 = 0 are parallel.

6) In the figure given below, let the lines l₁ and l₂ be parallel and m is transversal. If <F = 65°, find the measure of each of the remaining angles.

7) In the figure given below, let the lines l₁ and l₂ be parallel and t is transversal. Find the value of "x"

8) In the figure given below, let the lines l₁ and l₂ be parallel and t is transversal. Find the value of "x"

9) The slopes of the two lines are 7 and (3k +2). If the two lines are perpendicular, find the value of "k"

10) The equations of the two perpendicular lines are

3x + 2y - 8 = 0

(5k+3) - 3y + 1 = 0

Find the value of "k"

11) Find the equation of a straight line is passing through (2, 3) and perpendicular to the line 2x - y + 7 = 0.

12) Verify, whether the two lines 3x - 2y - 7 = 0 and y = - (2x/3) + 4 are perpendicular.

13) Verify, whether the two lines 5x + 7y - 1 = 0 and 14x - 10y + 5 = 0 are perpendicular.

**Problem 1 :**

The slopes of the two lines are 7 and (3k +2). If the two lines are parallel, find the value of "k"

**Solution :**

If the given two lines are parallel, then their slopes are equal.

3k + 2 = 7

3k = 5

**k = 5/3**

Let us look at the next problem on "Properties of parallel and perpendicular lines worksheet"

**Problem 2 :**

The equations of the two parallel lines are

3x + 2y - 8 = 0

(5k+3) + 2y + 1 = 0

Find the value of "k"

**Solution :**

If the two lines are parallel, then their general form of equations will differ only in the constant term and they will have the same coefficients of "x" and "y".

If the given two lines are parallel, then their slopes are equal.

3k + 2 = 7

3k = 5

**k = 5/3**

Let us look at the next problem on "Properties of parallel and perpendicular lines worksheet"

**Problem 3 :**

Find the equation of a straight line is passing through (2, 3) and parallel to the line 2x - y + 7 = 0.

**Solution :**

Since the required line is parallel to 2x - y + 7 = 0, the equation of the required line and the equation of the given line 2x - y + 7 = 0 will differ only in the constant term.

Therefore, equation of the required line is 2x -y + k = 0 ------> (1)

The required line is passing through (2, 3).

So, we can plug x = 2 and y = 3 in the equation of the required line.

2(2) - 3 + k = 0

4 - 3 + k = 0

1 + k = 0

k = - 1

**Hence, the equation of the required line is 2x -y - 1 = 0.**

Let us look at the next problem on "Properties of parallel and perpendicular lines worksheet"

**Problem 4 :**

Verify, whether the two lines 3x + 2y - 7 = 0 and y = -1.5x + 4 are parallel.

**Solution :**

In the equations of the given two lines, the equation of the second line is not in general form.

Let us write the equation of the second line in general form.

y = -1.5x + 4

1.5x + y - 4 = 0

Multiply by 2 on both sides,

3x + 2y - 8 = 0

Now, let us compare the equations of two lines,

3x + 2y - 7 = 0

3x + 2y - 8 = 0

The above two equations differ only in the constant term.

**Hence, the equations of the given two lines are parallel.**

Let us look at the next problem on "Properties of parallel and perpendicular lines worksheet"

**Problem 5 :**

Verify, whether the two lines 5x + 7y - 1 = 0 and 10x + 14y + 5 = 0 are parallel.

**Solution :**

In the equation of the second line 10x+14y + 5 = 0, the coefficients of "x" and "y" have the common divisor 2.

So, let us divide the second equation by 2

(10x/2) + (14y/2) + (5/2) = (0/2)

5x + 7y + 2.5 = 0

Now, let us compare the equations of two lines,

5x + 7y - 1 = 0

5x + 7y + 2.5 = 0

The above two equations differ only in the constant term.

**Hence, the equations of the given two lines are parallel.**

Let us look at the next problem on "Properties of parallel and perpendicular lines worksheet"

**Problem 6 :**

In the figure given below, let the lines l₁ and l₂ be parallel and m is transversal. If <F = 65°, find the measure of each of the remaining angles.

**Solution : **

From the given figure,

< F and < H are vertically opposite angles and they are equal.

Then, < H = < F -------> < H = 65°

< H and < D are corresponding angles and they are equal.

Then, < D = < H -------> < D = 65°

< D and < B are vertically opposite angles and they are equal.

Then, < B = < D -------> < B = 65°

< F and < E are together form a straight angle.

Then, we have

< F + < E = 180°

Plug < F = 65°

< F + < E = 180°

65° + < E = 180°

< E = 115°

< E and < G are vertically opposite angles and they are equal.

Then, < G = < E -------> < G = 115°

< G and < C are corresponding angles and they are equal.

Then, < C = < G -------> < C = 115°

< C and < A are vertically opposite angles and they are equal.

Then, < A = < C -------> < A = 115°

**Therefore, **

**< A = < C = < E = < G = 115°**

**< B = < D = < F = < H = 65°**

Let us look at the next problem on "Properties of parallel and perpendicular lines worksheet"

**Problem 7 :**

In the figure given below, let the lines l₁ and l₂ be parallel and t is transversal. Find the value of "x"

**Solution : **

From the given figure,

< (2x + 20)° and < (3x - 10)° are corresponding angles.

So, they are equal.

Then, we have

2x + 20 = 3x - 10

30 = x

**Hence, x = 30°**

Let us look at the next problem on "Properties of parallel and perpendicular lines worksheet"

**Problem 8 :**

In the figure given below, let the lines l₁ and l₂ be parallel and t is transversal. Find trhe value of "x"

**Solution : **

From the given figure,

< (3x + 20)° and < 2x° are consecutive interior angles.

So, they are supplementary.

Then, we have

3x + 20 + 2x = 180°

5x + 20 = 180°

5x = 160°

x = 32°

**Hence, x = 32°**

Let us look at the next problem on "Properties of parallel and perpendicular lines worksheet"

**Problem 9 :**

The slopes of the two lines are 7 and (3k +2). If the two lines are perpendicular, find the value of "k"

**Solution :**

If the given two lines are parallel, then the product of the slopes is equal to -1.

7(3k + 2) = - 1

21k + 14 = -1

21k = -15

k = -15/21

**k = -5/7**

Let us look at the next problem on "Properties of parallel and perpendicular lines worksheet"

**Problem 10 :**

The equations of the two perpendicular lines are

3x + 2y - 8 = 0

(5k+3) - 3y + 1 = 0

Find the value of "k"

**Solution :**

If the two lines are perpendicular, then the coefficient "y" term in the first line is equal to the coefficient of "x" term in the second line.

So, we have

5k + 3 = 2

5k = -1

**k = -1/5**

Let us look at the next problem on "Properties of parallel and perpendicular lines worksheet"

**Problem 11 :**

Find the equation of a straight line is passing through (2, 3) and perpendicular to the line 2x - y + 7 = 0.

**Solution :**

Since the required line is perpendicular to 2x - y + 7 = 0,

then, equation of the required line is x + 2y + k = 0 ------> (1)

The required line is passing through (2, 3).

So, we can plug x = 2 and y = 3 in the equation of the required line.

2 + 2(3) + k = 0

2 + 6 + k = 0

8 + k = 0

k = - 8

**Hence, the equation of the required line is x + 2y - 8 = 0.**

Let us look at the next problem on "Properties of parallel and perpendicular lines worksheet"

**Problem 12 :**

Verify, whether the two lines 3x - 2y - 7 = 0 and y = - (2x/3) + 4 are perpendicular.

**Solution :**

In the equations of the given two lines, the equation of the second line is not in general form.

Let us write the equation of the second line in general form.

y = - (2x/3) + 4

Multiply by 3 on both sides,

3y = - 2x + 12

2x + 3y - 12 = 0

Now, let us compare the equations of two lines,

3x - 2y - 7 = 0

2x + 3y - 12 = 0

When we look at the general form of equations of the above two lines, we get the following points.

(i) The sign of y- terms are different.

(ii) The coefficient of "x" term in the first equation is the coefficient of "y" term in the second equation.

(iii) The coefficient of "y" term in the first equation is the coefficient of "x" term in the second equation.

(iv) The above equations differ in constant terms.

Considering the above points, it is clear that the given two lines are perpendicular.

**Hence, the equations of the given two lines are perpendicular.**

Let us look at the next problem on "Properties of parallel and perpendicular lines worksheet"

**Problem 13 :**

Verify, whether the two lines 5x + 7y - 1 = 0 and 14x - 10y + 5 = 0 are perpendicular.

**Solution :**

In the equation of the second line 14x-10y + 5 = 0, the coefficients of "x" and "y" have the common divisor 2.

So, let us divide the second equation by 2

(14x/2) - (10y/2) + (5/2) = (0/2)

7x - 5y + 2.5 = 0

Now, let us compare the equations of two lines,

5x + 7y - 1 = 0

7x - 5y + 2.5 = 0

When we look at the general form of equations of the above two lines, we get the following points.

(i) The sign of y- terms are different.

(ii) The coefficient of "x" term in the first equation is the coefficient of "y" term in the second equation.

(iii) The coefficient of "y" term in the first equation is the coefficient of "x" term in the second equation.

(iv) The above equations differ in constant terms.

Considering the above points, it is clear that the given two lines are perpendicular.

**Hence, the equations of the given two lines are perpendicular.**

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