Parallel lines cut by a transversal solving equations worksheet answers

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When two parallel lines are “cut” by a transversal, some special properties arise.  We will begin by stating these properties, and then we can use these properties to solve some problems.

PROPERTY 1:  When two parallel lines are cut by a transversal, then corresponding angles are congruent.

In the diagram below, angles 1 and 5 are corresponding, and so they are equal.  Similarly with angles 4 and 8, etc.

PROPERTY 2:  When two parallel lines are cut by a transversal, then adjacent angles are supplementary.  That is, when two parallel lines are cut by a transversal, then the sum of adjacent angles is \(180^\circ \).

In the diagram above, this property tells us that angles 1 and 2 sum to \(180^\circ \).  Similarly with angles 5 and 6.

EXAMPLE:  Solve for \(x\).

SOLUTION:  From property 2, we know that the 2 angles \(x + 75\), and \(A\) (so called by me) are supplementary.  That is,

\(x + 75 + A = 180\)

But from property 1, we know that \(A = x + 125\), since those two angles are corresponding.  Then \(x + 75 + x + 125 = 180\), so that \(2x + 200 = 180\). Then \(2x =  - 20\), and \(x =  - 10\).

EXAMPLE:  Solve for \(x\).

SOLUTION:  Since, by property 1, we know that corresponding angles are congruent, we know that

\(12x + 3 = 11x + 9\)

\(x = 6\)

Below you can download some free math worksheets and practice.

Parallel lines and transversals worksheets can help students identify the different types of angles that can be formed like corresponding angles, vertical angles, alternate interior angles, alternate exterior angles. They can utilize this knowledge of the angles formed by parallel lines and transversals to set up and solve equations for missing angles.

Benefits of Parallel Lines and Transversals Worksheets

Parallel lines and transversals worksheets can help students to learn about angles formed by parallel lines cut by a transversal. Parallel lines and transversals worksheets will help kids in solving geometry problems. Some real-life examples of parallel lines cut by a transversal are zebra crossing on the road, road and railway crossing, railway tracks with sleepers, and windscreen wipers in cars.

Printable PDFs for Parallel Lines and Transversals Worksheets

First, students will need to be able to identify angle pairs, then know the properties and relationships. Children and parents can find these math worksheets online or even download the PDF format of these exciting worksheets.

A transversal is a line that intersects two lines in the same plane at two different points.

In the diagram shown below, let the lines 'a' and 'b' be parallel. Because the line 't' cuts the lines 'a' and 'b', the line 't' is transversal.

So, the two parallel lines 'a' and 'b' cut by the transversal 't'.

We can have the following important results from the above diagram.

Corresponding Angles :

Angles lie on the same side of the transversal t, on the same side of lines a and b.

Example : ∠ 1 and ∠ 5

Alternate Interior Angles :

Angles are nonadjacent angles that lie on opposite sides of the transversal t, between lines a and b.

Example : ∠ 3 and ∠ 6

Alternate Exterior Angles :

Angles lie on opposite sides of the transversal t, outside lines a and b.

Example : ∠ 1 and ∠ 8

Same-Side Interior Angles :

Angles lie on the same side of the transversal t, between lines a and b.

Example : ∠ 3 and ∠ 5

Reflect

Identify the pairs of angles in the diagram. Then make a conjecture about their angle measures.

Corresponding Angles : 

∠CGE and ∠AHG, ∠DGE and ∠BHG, ∠CGH and ∠AHF, ∠DGH and ∠BHF ; congruent.

Alternate Interior Angles :

∠CGH and ∠BHG, ∠DGH and ∠AHG ; congruent.

Alternate Exterior Angles :

∠CGE and ∠BHF, ∠DGE and ∠AHF ; congruent.

Same-Side Interior Angles :

∠CGH and ∠AHG, ∠DGH and ∠BHG ; supplementary.

Solved Problems

Problem 1 :

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°

Problem 2 :

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)°

2x + 20  =  3x - 10

Subtract 2x from each side. 

20  =  x - 10

Add 10 to each side. 

30  =  x

Problem 3 :

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, 

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

So, they are supplementary. 

Then, we have

(3x + 20)° + 2x°  =  180°

3x + 20 + 2x  =  180

Simplify.

5x + 20  =  180

Subtract 20 from each side.

5x  =  160

Divide each side by 8.

x  =  32

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