Resources For Teachers For Tutors For Students & Parents AI New Pricing
Create Free Account
Year 7 Maths Angles

Co-interior Angles on Parallel Lines

20 practice questions 1 video lesson Theory + worked examples
Practice 20 questions
Practice questions

Every question with a fully worked solution.

Start practising
Watch 1 video(s)
  • Co-interior Angles on Parallel Lines - Video - Angles in parallel lines- co-interior angles Watch
Create a free accountTrack your progress and save your work as you go.
Create free account

Theory

Co-interior angles are two angles between a pair of lines, on the same side of a transversal. When the lines are parallel, co-interior angles add to \(180^\circ\). This Year 7 maths page shows how to spot them (a C shape) and use them with reasons.

A transversal is a line that crosses two or more other lines. Arrow marks show that lines are parallel, written \(AB \parallel CD\).

Co-interior angles sit between the two lines, on the same side of the transversal. Together with the lines they make a C shape (it may be back to front).

On parallel lines, co-interior angles are supplementary: they add to \(180^\circ\).

To find an unknown angle, write the reason, then the equation: co-interior angles, \(AB \parallel CD\): \(a + 111 = 180\), so \(a = 69\).

Co-interior angles on the right Parallel lines cut by a transversal. Between the lines on the right of the transversal: 113 degrees at the top crossing and 67 degrees at the bottom crossing. 113° 67°
Right of the transversal: a C shape. \(113^\circ + 67^\circ = 180^\circ\).
Co-interior angles on the left Parallel lines cut by a transversal. Between the lines on the left: 67 degrees at the top crossing and 113 degrees at the bottom crossing. 67° 113°
Left of the transversal: a back-to-front C. \(67^\circ + 113^\circ = 180^\circ\). In each pair one angle is acute and one is obtuse (unless both are \(90^\circ\)).

The rule:

Co-interior angles on parallel lines add to \(180^\circ\).

\[ a + b = 180^\circ \]
a + b =180
A point between the lines. Draw a line through it, parallel to both lines. Each part of the angle at the point then makes a co-interior pair with one of the lines.
Trapeziums. In a trapezium with \(AB \parallel DC\), each slanting side is a transversal, so the two angles on that side add to \(180^\circ\).

How to use co-interior angles

  1. Check that the lines are parallel.
  2. Check both angles are between the lines, on the same side.
  3. Write the reason and the equation: the two angles add to \(180\).
  4. Solve, and check that one angle is acute and one obtuse.
Example 1 — Find a co-interior angle
\(AB \parallel CD\). Find the value of \(a\).
124 and a Parallel lines AB and CD. 124 degrees below AB on the right; a above CD on the right. 124° a° A B C D
Solution

Both angles are between the lines, on the right: co-interior angles, \(AB \parallel CD\).

\(a + 124\)\(=\)\(180\)
\(a\)\(=\)\(56\)
a +124=180, a =56

\(a = 56\) (acute, as it looks)

Example 2 — Two steps with reasons
\(AB \parallel CD\) and \(\angle AGH = 59^\circ\). Find
(i) \(\angle GHC\)
(ii) \(\angle DHF\)
Angle AGH is 59 degrees Parallel lines AB and CD cut by EF at G and H. Angle AGH, below AB on the left, is 59 degrees. 59° A B C D E F G H
Solution
\(\angle GHC\)\(=\)\(180^\circ - 59^\circ = 121^\circ\)
\(\angle DHF\)\(=\)\(\angle GHC = 121^\circ\)
GHC =180−59=121, DHF = GHC =121

(i) Co-interior angles, \(AB \parallel CD\): \(121^\circ\)
(ii) Vertically opposite angles are equal: \(121^\circ\)

Example 3 — Vertically opposite, then co-interior
\(AB \parallel CD\) and \(\angle CGH = 68^\circ\). Find the size of \(\angle BFG\).
Angle CGH is 68 degrees Parallel lines AB and CD cut by EH at F and G. Angle CGH, below CD on the left, is 68 degrees. 68° A B C D E H F G
Solution
\(\angle FGD\)\(=\)\(\angle CGH = 68^\circ\)
\(\angle BFG\)\(=\)\(180^\circ - 68^\circ = 112^\circ\)
FGD = CGH =68, BFG =180−68=112

Vertically opposite angles are equal; then co-interior angles, \(AB \parallel CD\). \(\angle BFG = 112^\circ\).

Example 4 — A straight line, then co-interior
\(PQ \parallel RS\) and \(\angle RVW = 104^\circ\). Find the size of \(\angle PUV\).
Angle RVW is 104 degrees Parallel lines PQ and RS cut by TW at U and V. Angle RVW, below RS on the left, is 104 degrees. 104° P Q R S T W U V
Solution
\(\angle RVU\)\(=\)\(180^\circ - 104^\circ = 76^\circ\)
\(\angle PUV\)\(=\)\(180^\circ - 76^\circ = 104^\circ\)
RVU =180−104=76, PUV =180−76=104

Angles on a straight line add to \(180^\circ\); then co-interior angles, \(PQ \parallel RS\). \(\angle PUV = 104^\circ\).

Example 5 — Angles in a trapezium
\(ABCD\) is a trapezium with \(AB \parallel DC\). Find \(a\) and \(b\).
Trapezium ABCD Trapezium ABCD with AB parallel to DC. Angle D is 74 degrees, angle C is 101 degrees, angle A is a and angle B is b. 74° 101° a° b° A B C D
Solution

Side \(AD\) is a transversal, and so is side \(BC\). Co-interior angles, \(AB \parallel DC\).

\(a + 74\)\(=\)\(180\)
\(a\)\(=\)\(106\)
\(b + 101\)\(=\)\(180\)
\(b\)\(=\)\(79\)
a +74=180, a =106 b +101=180, b =79

\(a = 106\), \(b = 79\)

Example 6 — Write and solve an equation
\(AB \parallel CD\). Find \(x\), then the size of each angle.
2x + 15 and x Parallel lines AB and CD. (2x + 15) degrees below AB on the right; x degrees above CD on the right. (2x+15)° x° A B C D
Solution

Both angles are between the lines, on the right: co-interior angles, \(AB \parallel CD\).

\(x + (2x + 15)\)\(=\)\(180\)
\(3x\)\(=\)\(165\)
\(x\)\(=\)\(55\)
x +2 x +15=180,3 x =165, x =55

\(x = 55\). The angles are \(55^\circ\) and \(2 \times 55 + 15 = 125^\circ\). Check: \(55 + 125 = 180\).

Common pitfalls

Co-interior angles are not equal. They add to \(180^\circ\). If one is \(67^\circ\), the other is \(113^\circ\).
An angle outside the lines. Both angles must be between the lines.
No arrow marks. The rule works only when the lines are parallel. Write the reason with the lines named: co-interior angles, \(AB \parallel CD\).
Two acute (or two obtuse) answers. On parallel lines one co-interior angle is acute and the other obtuse, unless both are \(90^\circ\).

Frequently asked questions

What are co-interior angles?

Two angles between a pair of lines, on the same side of a transversal.

What do co-interior angles on parallel lines add to?

\(180^\circ\).

Are co-interior angles equal?

No (unless both are \(90^\circ\)). They are supplementary.

What shape do co-interior angles make?

A C shape, which may be back to front.

How are co-interior angles used in a trapezium?

Each slanting side is a transversal, so the angles at its ends add to \(180^\circ\).

More in Angles
Naming Angles
20 questions · Video
Measuring Angles
20 questions · Video
Classifying Angles
20 questions · Video
Complementary and Supplementary Angles
20 questions · Video
Angles at a Point and Vertically Opposite Angles
20 questions · Video
Parallel and Perpendicular Lines
20 questions · Video
Corresponding Angles on Parallel Lines
20 questions · Video
Alternate Angles on Parallel Lines
20 questions · Video
Angles on Parallel Lines
20 questions · Video
Proving Parallel Lines
20 questions · Video