Angles in Parallel Lines
Angles in parallel lines are the three special angle relationships created when a transversal line crosses a pair of parallel lines: corresponding angles, in an F-shape, are equal, alternate angles, in a Z-shape, are equal, and co-interior angles, also called allied angles, in a C-shape, add up to 180 degrees. These facts, combined with angles on a straight line and vertically opposite angles, let any missing angle in a parallel lines diagram be found.
Before you start
Make sure you're comfortable with these topics first:
Method
- Identify the parallel lines (usually shown with matching arrow marks) and the transversal line that crosses them.
- Look for an F-shape in the diagram: corresponding angles are equal.
- Look for a Z-shape in the diagram: alternate angles are equal.
- Look for a C-shape (or U-shape) in the diagram: co-interior (allied) angles add up to 180 degrees.
- Combine these facts with angles on a straight line (180 degrees) or vertically opposite angles (equal) if the angle you need is not directly one of the three types above.
- State your answer with the angle fact used as a reason, for example 'alternate angles are equal' or 'co-interior angles sum to 180 degrees'.
Worked example
Two parallel lines are crossed by a transversal. One of the angles formed is 72 degrees. Find (a) its corresponding angle, (b) its alternate angle, (c) its co-interior angle.
- Corresponding angles are equal, so the corresponding angle is also 72 degrees.
- Alternate angles are equal, so the alternate angle is also 72 degrees.
- Co-interior angles sum to 180 degrees, so the co-interior angle is 180 - 72 = 108 degrees.
Practice questions
Try each question, then tap to reveal the answer.
Q1Two parallel lines are cut by a transversal. One angle is 65 degrees. State the size of its corresponding angle.Show answer
Answer: 65 degrees, since corresponding angles are equal.
Q2Using the same transversal, state the size of the angle that is alternate to a 65 degree angle.Show answer
Answer: 65 degrees, since alternate angles are equal.
Q3State the size of the co-interior angle to a 65 degree angle.Show answer
Answer: 180 - 65 = 115 degrees
Q4Two angles are vertically opposite and one is 48 degrees. Find the other.Show answer
Answer: 48 degrees, since vertically opposite angles are equal.
Q5Explain what an F-shape in a diagram tells you about two angles.Show answer
Answer: The two angles are corresponding angles, so they are equal.
Q6Two co-interior angles are 3x and (2x + 30). Find x.Show answer
Answer: 3x + (2x + 30) = 180, so 5x + 30 = 180, 5x = 150, and x = 30
Q7An angle of 110 degrees and an angle of x are alternate angles. Find x.Show answer
Answer: x = 110 degrees, since alternate angles are equal.
Q8Two parallel lines are cut by a transversal, forming a Z-shape between an angle of (4x - 10) degrees and an angle of (2x + 30) degrees. Find x.Show answer
Answer: Alternate angles are equal, so 4x - 10 = 2x + 30, giving 2x = 40 and x = 20
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
Two parallel lines are cut by a transversal. One angle is labelled (2x + 15) degrees and its co-interior angle is labelled (3x - 10) degrees. Work out the value of x, and state the size of each angle.
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Line AB is parallel to line CD. A transversal crosses both lines. The angle between the transversal and AB is (x + 40) degrees, and the alternate angle between the transversal and CD is (2x - 10) degrees. (a) Find the value of x. (b) State the size of each of these two angles, and explain why they are equal.
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Free printable worksheet
Want more practice on paper? Download the angles in parallel lines worksheet pack - 8 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 9 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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