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Angles in a Quadrilateral

FoundationChallengeAQAEdexcelOCR

Get to grips with angles in a quadrilateral using these Foundation GCSE Maths practice questions. The worksheet focuses on finding angles in a quadrilateral, and the accompanying mark scheme breaks down each solution clearly. Suitable for AQA, Edexcel and OCR. Download the questions and answers for free. The angles in a quadrilateral add up to 360°.

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Topic overview

The four angles inside any quadrilateral add to \(360^\circ\). The reason is that a quadrilateral can be split into two triangles by drawing one diagonal, and each triangle contributes \(180^\circ\).

Special quadrilaterals give more information. A square and a rectangle have four right angles. A parallelogram has opposite angles equal and adjacent angles adding to \(180^\circ\).

A kite has one pair of equal angles, between the unequal sides, and a rhombus behaves like a parallelogram with all sides equal. Identifying the shape correctly usually gives you enough extra facts to finish the question.

Revision notes

The angle sum

The four interior angles of any quadrilateral add to \(360^\circ\).

If three angles are \(80^\circ\), \(95^\circ\) and \(110^\circ\), the fourth is \(360 - 285 = 75^\circ\).

Parallelograms and rhombuses

Opposite angles are equal, and adjacent angles add to \(180^\circ\) because they lie between parallel lines.

So if one angle of a parallelogram is \(70^\circ\), the opposite angle is also \(70^\circ\) and the other two are \(110^\circ\) each.

Why the sum is 360

Drawing a diagonal splits the quadrilateral into two triangles.

Each triangle contributes \(180^\circ\), so together they give \(360^\circ\). This is worth knowing as an explanation, since questions sometimes ask you to justify the rule.

Key points

  • Angles in a quadrilateral add to \(360^\circ\).
  • A diagonal splits it into two triangles.
  • Squares and rectangles have four right angles.
  • Opposite angles of a parallelogram are equal.
  • Adjacent angles of a parallelogram add to \(180^\circ\).
  • Identify the shape to gain extra facts.

Worked examples

Example 1

Three angles of a quadrilateral are \(85^\circ\), \(100^\circ\) and \(70^\circ\). Find the fourth.

Working

\[85 + 100 + 70 = 255\]add the three known angles
\[360 - 255\]angles in a quadrilateral add to 360 degrees
\[= 105^\circ\]state the fourth angle

Example 2

One angle of a parallelogram is \(65^\circ\). Find the other three.

Working

\[65^\circ \text{ opposite}\]opposite angles are equal
\[180 - 65 = 115^\circ\]adjacent angles add to 180 degrees
\[65, 115, 65, 115\]list all four angles

Example 3

Explain why the angles of a quadrilateral add to \(360^\circ\).

Working

\[\text{A diagonal splits it into two triangles}\]draw one diagonal
\[2 \times 180 = 360^\circ\]each triangle contributes 180 degrees

Common mistakes

  • Using 180 instead of 360.

    A quadrilateral has four angles summing to 360, not 180.

  • Assuming all quadrilaterals have equal opposite angles.

    That holds for parallelograms and rhombuses, not for every quadrilateral.

  • Misidentifying the shape.

    Check the markings for equal sides and parallel arrows before applying extra facts.

  • Forgetting to give a reason.

    State the angle sum fact for the reasoning mark.

Exam tips

  • Identify the type of quadrilateral first.
  • Look for arrows marking parallel sides.
  • Use the two-triangle explanation if asked to justify the rule.
  • Check your four angles add to 360 before finishing.

Key terms

Quadrilateral
A shape with four straight sides.
Parallelogram
A quadrilateral with two pairs of parallel sides.
Diagonal
A line joining opposite corners.
Adjacent
Next to each other.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.