Intersecting and parallel lines

Part ofMathsAngles

Key points

An image of three horizontal, parallel lines. Each line is marked by a single arrow, which points right, to indicate the lines are parallel. The top line is coloured blue, the middle line is coloured orange, and the bottom line is coloured purple.
Parallel lines are indicated by arrows going in the same direction.

  • Lines that cross each other are called . The point at which they cross is known as the .

    • If two lines meet at a , they are called lines.
    • At the point of intersection, the vertically opposite angles are equal.

  • Straight lines that remain the same distance apart and never intersect are known as .

  • If a line passes through two or more parallel lines, it is known as a . A transversal intersection forms various types of angles with the parallel lines:

    • on parallel lines are always equal.
    • on parallel lines are always equal.

  • It is useful to remember that angles on a straight line add up to 180° and that angles in a triangle add up to 180°.

An image of three horizontal, parallel lines. Each line is marked by a single arrow, which points right, to indicate the lines are parallel. The top line is coloured blue, the middle line is coloured orange, and the bottom line is coloured purple.
Parallel lines are indicated by arrows going in the same direction.
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Understanding intersecting and parallel lines

  • Two or more lines which cross at a point are referred to as .

  • Two or more lines which remain the same distance apart and never touch are referred to as .

    • Parallel lines are indicated by arrows going in the same direction.

    • If there is more than one set of parallel lines in a diagram, different numbers of arrows are used.

Examples

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  1. An image of three straight lines. The three straight lines all cross at a single point. Written above: Point of intersection. An arrow point from the text to the point where the lines intersect. One line is coloured blue, one line is coloured orange, and one line is coloured purple.

    The three lines in this image are intersecting. They intersect at a common point. This is called the point of intersection.

Question

How many pairs of parallel lines are shown in this diagram?

An image of six straight lines. Two lines are both marked with a single arrow to indicate the lines are parallel. Two lines are both marked with two arrows to indicate the lines are parallel. Two lines are both marked with three arrows to indicate the lines are parallel.

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Angles in intersecting lines

  • When two lines intersect, the opposite angles are equal.

  • The opposite angles that are found at the are known as .

Example

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  1. A series of two images. The first image shows two diagonal, intersecting lines. The angle to the left and right of the point of intersection are both coloured blue. The second image shows the same two diagonal, intersecting lines. The angle above and below of the point of intersection are both coloured orange.

    In these pairs of intersecting lines, the opposite angles are equal. They are referred to as vertically opposite angles.

Question

Work out the size of angles 𝒙, 𝒚 and 𝒛.

An image of three straight lines. Two of the lines intersect to the left. The angle above this point of intersection has been labelled as one hundred and forty degrees. The angle below this point of intersection has been labelled x. Two of the lines intersect to the right. The angle above this point of intersection has been labelled as seventy degrees. The angle to the right of this point of intersection has been labelled y. The angle to the left of this point of intersection has been labelled z. Angle x is coloured orange, angle y is coloured blue and angle z is coloured purple.

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Angles in parallel lines

When a intersects a pair of parallel lines, the angles at both points of intersection are related. These angles can be found using different connections. One of these connections is .

  • Alternate angles are on opposite sides of the transversal within the parallel lines.
  • Alternate angles are always equal in size.
  • When looking for alternate angles, it can be helpful to look for a Z-shape.
    • The Z-shape can be backwards, sideways or upside down.

Another of these connections is .

  • Corresponding angles occur at the same position within each intersection.
  • Corresponding angles are always equal in size.
  • When looking for corresponding angles, it can be helpful to look for an F-shape.
    • The F-shape can be backwards, sideways or upside down.

Examples

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  1. A series of three images. The first image has three straight lines. Two lines are horizontal and are both marked with a single arrow to indicate the lines are parallel. A diagonal line, or transversal, starting in the bottom left and ending in the top right intersects both the horizontal lines. Where the diagonal line intersects the top horizontal line, the angle in the bottom left has been marked, and where the diagonal line intersects the bottom horizontal line, the angle in the top right has been marked. The arms of the angles have been highlighted to make a Z shape. The Z shape is coloured orange. The second image has three straight lines. Two lines are horizontal and are both marked with a single arrow to indicate the lines are parallel. A diagonal line, or transversal, starting in the bottom left and ending in the top right intersects both the horizontal lines. Where the diagonal line intersects the top horizontal line, the angle in the bottom right has been marked, and where the diagonal line intersects the bottom horizontal line, the angle in the top left has been marked. The arms of the angles have been highlighted to make a stretched Z shape. The Z shape is coloured blue. The third image has three straight lines. Two lines are diagonal and are both marked with a single arrow to indicate the lines are parallel. The diagonal lines start in the bottom left and end in the top right. A horizontal line passes through the diagonal lines intersecting both. Where the horizontal line intersects the left diagonal line, the angle in the top right has been marked, and where the horizontal line intersects the right diagonal line, the angle in the bottom left has been marked. The arms of the angles have been highlighted to make a backwards Z shape. The Z shape is coloured pink.

    These images all show alternate angles. The angles on opposite sides of the transversal within the parallel lines are always equal.

Questions

Question 1: Which angle is alternate to 130°?

The image has three straight lines. Two lines are horizontal and are both marked with a single arrow to indicate the lines are parallel. A diagonal line, or transversal, starting in the bottom left and ending in the top right intersects both the horizontal lines. Where the diagonal line intersects the top horizontal line, the angle in the bottom left has been labelled a. Where the diagonal line intersects the top horizontal line, the angle in the bottom right has been labelled b. Where the diagonal line intersects the bottom horizontal line, the angle in the top left has been labelled as one hundred and thirty degrees. Where the diagonal line intersects the bottom horizontal line, the angle in the top right has been labelled as c. The letter a is coloured orange, the letter b is coloured blue, and the letter c is coloured purple.

Question 2: Which angle is corresponding to 130°?

The image has three straight lines. Two lines are horizontal and are both marked with a single arrow to indicate the lines are parallel. A diagonal line, or transversal, starting in the bottom left and ending in the top right intersects both the horizontal lines. Where the diagonal line intersects the top horizontal line, the angle in the top left has been labelled a. Where the diagonal line intersects the top horizontal line, the angle in the bottom right has been labelled b. Where the diagonal line intersects the bottom horizontal line, the angle in the top left has been labelled as one hundred and thirty degrees. Where the diagonal line intersects the bottom horizontal line, the angle in the top right has been labelled as c. The letter a is coloured orange, the letter b is coloured blue, and the letter c is coloured purple.

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Practise finding intersecting and parallel lines

Practise finding intersecting and parallel lines with this quiz. You may need a pen and paper to help you with your answers.

Quiz

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Real-life maths

An architect's hands measuring a distance on a floor plan.
To make sure buildings are safe and stable, architects use angles in parallel lines.

Angles in parallel lines are particularly useful in architecture.

Architects use angles in parallel lines to make sure buildings are stable. If the top and bottom of a building are parallel, the building should not topple. This can be checked by confirming that the corresponding, or alternate, angles at the top and bottom are equal.

An architect's hands measuring a distance on a floor plan.
To make sure buildings are safe and stable, architects use angles in parallel lines.
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