Solve Problems Involving Parallel Lines Worksheet

Parallel lines have so much in common - it's a shame they will never meet!

parallel lines

Engineers, architects and designers all need to know and use key facts about angles and straight lines in their work - we wouldn't want a wonky house or an aircraft with uneven wings, would we?

Knowing the properties (facts) of angles and parallel lines can help us to solve all sorts of problems.

Let's remind ourselves of the key facts now...

Diagram of angles around a point

Angles on a straight line add up to 180°:

Diagram showing examples of interior angles

Alternate angles (or Z-angles) are equal:

Diagram showing example of alternate angles

Corresponding angles (or F-angles) are also equal:

Diagram showing examples of corresponding angles

Often we will see more than one of these rules in action within the same problem, so we need to have our eagle eyes ready to spot these scenarios.

Lets give this a go now in some examples.

e.g. Find the value of angle x in the diagram below:

Diagram showing angles between two parallel lines with one missing

Here, we need to draw another parallel line through the centre of our diagram to help.

Don't worry, there are no rules to say that we can't do this!

Diagram showing angles between two parallel lines with one missing

This has split our target angle (x) into two elements.

We can now apply the alternate angle rule from above to link these two elements to the angles already provided:

x = 28° + 38° = 66°

e.g. Find the value of x and y in the diagram below:

Diagram showing angles between two parallel lines with two missing

Angle y is a corresponding angle to 114° so these must have the same value.

Angle x and y are on a straight line, so together will add up to 180°.

Therefore, x = 180° - 114° = 66°

Now we are going to put all the rules we know relating to angles and straight lines together to solve problems which involve taking multiple steps to calculate unknown angles using number and algebra.

Review the known and unknown angles on this new diagram:

Diagram showing angles between parallel lines

What is the value of angle x?

92°

48°

46°

88°

Investigate the known and unknown angles on this final diagram now:

Diagram showing angles between parallel lines using algebra

What is the value of x?

177°

27°

130°

59°

Observe the known and unknown angles on the diagram below:

Diagram showing angles between parallel lines

Then select the correct option to complete the sentence below.

The value of angle x is 87° / 93° / 77°.

ANSWERS

Question 1

Review the known and unknown angles on this new diagram:

Diagram showing angles between parallel lines

What is the value of angle x?

CORRECT ANSWER

88°

EDDIE SAYS

This question is similar to the example in our Introduction - we need to draw an extra parallel line this time. Once we do this, we can then apply the rule related to alternate angles to find the two elements which combine to form missing angle x. The lower part of angle x is alternate with 46°, whilst the upper part is alternate with 42°. So x = 46° + 42° = 88°

ANSWERS

Question 2

Next, review the known and unknown angles on this new diagram:

Diagram showing angles between parallel lines

Then select the correct answers to complete the expressions below.

EDDIE SAYS

Wow, we had a lot of angles to find here! Let's just take them one at a time... Angle x is alternate to 40° so these angles are the same. The value of angle y can be calculated using the rule related to angles on a straight line, as we are provided with its neighbouring angle: 180° - 46° = 134° Angle z can be calculated in one of two ways. You may have spotted that it is corresponding to angle y, or that interior angles add up to 180°: 180° - 46° = 134° And the beauty here is that we can work things out in any order we choose - what a treat!

ANSWERS

Question 3

Explore the known and unknown angles on the diagram below:

Diagram showing angles between parallel lines

Now type numbers to complete the statements below.

CORRECT ANSWER

EDDIE SAYS

Don't let the double set of parallel lines put you off here. Angle x is alternate to the angle labelled as 65°, whilst angle y is corresponding to 69°. No working out required this time, which is great, so long as we know, and can apply, the rules!

ANSWERS

Question 4

Investigate the known and unknown angles on the diagram below:

Diagram showing angles between parallel lines

Now type numbers to complete the statements below.

CORRECT ANSWER

EDDIE SAYS

Two sets of parallel lines again here for the price of one! Let's apply the rule that interior angles add up to 180° to find angle x: 180° - 62° = 118° We can apply this same rule again to find angle z: 180° - 118° = 62° Angle y is alternate to 100°, so these two angles are the same. Are you getting used to spotting different rules within the same diagram now?

ANSWERS

Question 5

Explore the known and unknown angles on the diagram below:

Diagram showing angles between parallel lines

What is the value of angle y°?

CORRECT ANSWER

EDDIE SAYS

Sometimes sneaky examiners put in an angle that we don't need just to try to confuse us and check that we really understand the rules. Could you spot the alternate angle here? Missing angle y is alternate to the angle shown as 64° - see how they make a Z-shape within our parallel lines? This means that these angles are the same, so: y = 64°

ANSWERS

Question 6

Explore the known and unknown angles on this next diagram now:

Diagram showing angles between parallel lines

Which two values represent angles x and y?

CORRECT ANSWER

EDDIE SAYS

The sneaky part here is that we need to find the angles we are not asked for first. Angle y is alternate to the angle labelled as 42°, so these two angles are the same. Look at the markings on the triangle now. They show that this triangle is an isosceles triangle. This means that the two base angles of the triangle are the same. So we need a base angle to work with in order to calculate angle x: 180° - 42° = 138° 138° ÷ 2 = 69° So angle x is on a straight line with 69°: 180° - 69° = 111°

ANSWERS

Question 7

Explore the angles on the diagram below which have been expressed algebraically:

Diagram showing angles between parallel lines

What are the values of x and y?

EDDIE SAYS

Here we have some algebra thrown in for good measure too! Before we do anything, we need to find the missing angles on the straight line with 2x and x. The missing angle is alternate to 72°. We know that there are '3 lots of x' (x + 2x) next to this angle, so the value of: 3x = 180° - 72° = 108° If we share this value between 3, we have the value of x: x = 108° ÷ 3 = 36° Angle x is alternate to angle y, so these two angles are the same.

ANSWERS

Question 8

Review the angles on the diagram below which have been expressed algebraically:

Diagram showing angles between parallel lines

What is the value of x?

CORRECT ANSWER

EDDIE SAYS

This time, we just need to use the parallel lines provided. The other base angle in the triangle is alternate to the angle labelled as 52°, so these are the same. We know that the angles in a triangle add up to 180°, so to find the value of angle x, we can subtract our two known angles from 180: 180° - 78° - 52° = 50°

ANSWERS

Question 9

Investigate the known and unknown angles on this final diagram now:

Diagram showing angles between parallel lines using algebra

What is the value of x?

CORRECT ANSWER

59°

EDDIE SAYS

Oh a final curve ball to deal with - no problem! We know that interior angles add up to 180° so: (2x - 12) + (x + 15) = 180° 3x + 3 = 180° 3x = 177° x = 59° We can put this value of x back into our original equation to check that it is correct and that our two angles will add up to 180°: 2x - 12 = (2 × 59) - 12 = 106° x + 15 = 59 + 15 = 74° 106° + 74° = 180° Great job! We have practised combining our knowledge of angle properties and parallel lines to solve problems which involve taking multiple steps to calculate unknown angles using number and algebra.

ANSWERS

Question 10

Observe the known and unknown angles on the diagram below:

Diagram showing angles between parallel lines

Then select the correct option to complete the sentence below.

CORRECT ANSWER

The value of angle x is 87° / 93° / 77°.

EDDIE SAYS

We need to work out other angles, that we have not been asked to find, first. The angle on the line next to missing angle x is corresponding to the angle labelled as 87°. Now we know that angles on a straight line add up to 180°, so we can find angle x by subtracting its neighbour from 180: 180° - 87° = 93°