# Solve Problems Involving Parallel Lines

In this worksheet, students will combine their knowledge of angle properties and parallel lines to solve problems which involve taking multiple steps to calculate unknown angles using number and algebra.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, AQA, Eduqas, OCR

Curriculum topic:   Geometry and Measures, Basic Geometry

Curriculum subtopic:   Properties and Constructions, Angles

Difficulty level:

### QUESTION 1 of 10

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

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...

Angles on a straight line add up to 180°:

Alternate angles (or Z-angles) are equal:

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

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:

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!

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:

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:

What is the value of angle x?

92°

48°

46°

88°

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

Then select the correct answers to complete the expressions below.

 134° 40° 180° The value of x is... The value of y is... The value of z is...

Explore the known and unknown angles on the diagram below:

Now type numbers to complete the statements below.

 134° 40° 180° The value of x is... The value of y is... The value of z is...

Investigate the known and unknown angles on the diagram below:

Now type numbers to complete the statements below.

 134° 40° 180° The value of x is... The value of y is... The value of z is...

Explore the known and unknown angles on the diagram below:

What is the value of angle ?

 134° 40° 180° The value of x is... The value of y is... The value of z is...

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

Which two values represent angles x and y?

 134° 40° 180° The value of x is... The value of y is... The value of z is...

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

What are the values of x and y?

 108° 72° 36° 110° The value of x is... The value of y is...

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

What is the value of x?

 108° 72° 36° 110° The value of x is... The value of y is...

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

What is the value of x?

177°

27°

130°

59°

Observe the known and unknown angles on the diagram below:

Then select the correct option to complete the sentence below.

The value of angle x is 87° / 93° / 77°.
• Question 1

Review the known and unknown angles on this new diagram:

What is the value of angle x?

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°
• Question 2

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

Then select the correct answers to complete the expressions below.

 134° 40° 180° The value of x is... The value of y is... The value of z is...
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!
• Question 3

Explore the known and unknown angles on the diagram below:

Now type numbers to complete the statements below.

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!
• Question 4

Investigate the known and unknown angles on the diagram below:

Now type numbers to complete the statements below.

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?
• Question 5

Explore the known and unknown angles on the diagram below:

What is the value of angle ?

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°
• Question 6

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

Which two values represent angles x and y?

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°
• Question 7

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

What are the values of x and y?

 108° 72° 36° 110° The value of x is... The value of y is...
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.
• Question 8

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

What is the value of x?

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°
• Question 9

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

What is the value of x?

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.
• Question 10

Observe the known and unknown angles on the diagram below:

Then select the correct option to complete the sentence below.

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°
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