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6.1 Properties of Shapes

GCSE Maths Foundation · Edexcel · Chapter 6: Angles

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: basic angle facts, the special quadrilaterals and the angle sum of a quadrilateral. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Use angles on a straight line, angles around a point and vertically opposite angles
  • Know the properties of the special quadrilaterals
  • Use the fact that the angles in a quadrilateral add up to 360°
  • Solve angle problems in quadrilaterals, including ones with algebra

Key words

  • Quadrilateral: A flat shape with 4 straight sides and 4 angles.
  • Parallel: Lines that stay the same distance apart and never meet. Shown with arrows.
  • Perpendicular: At right angles (90°) to each other.
  • Diagonal: A straight line joining two corners that are not next to each other.
  • Vertically opposite: The angles opposite each other where two straight lines cross.

Basic angle facts

  • Angles on a straight line; Rule add up to 180°
  • Angles around a point; Rule add up to 360°
  • Vertically opposite angles; Rule are equal
  • A right angle; Rule is 90° (shown with a small square)

Worked example: angles on a line and around a point

(a) Two angles on a straight line are 47° and x. x = 180° − 47° = 133°
(b) Angles around a point are 105°, 90°, 85° and y. y = 360° − (105° + 90° + 85°) = 360° − 280° = 80°

Answer: x = 133°, y = 80°

The special quadrilaterals

Dashes (ticks) show equal sides. Arrows show parallel sides. Small squares show right angles.

  • Square; Sides 4 equal; 2 pairs parallel; Angles all 90°; Diagonals equal; cross at 90°; bisect each other; Lines of symmetry 4
  • Rectangle; Sides opposite equal; 2 pairs parallel; Angles all 90°; Diagonals equal; bisect each other; Lines of symmetry 2
  • Parallelogram; Sides opposite equal; 2 pairs parallel; Angles opposite equal; Diagonals bisect each other; Lines of symmetry 0
  • Rhombus; Sides 4 equal; 2 pairs parallel; Angles opposite equal; Diagonals cross at 90°; bisect each other; Lines of symmetry 2
  • Kite; Sides 2 pairs of equal adjacent sides; Angles 1 pair equal; Diagonals cross at 90°; Lines of symmetry 1
  • Trapezium; Sides 1 pair parallel; Angles –; Diagonals –; Lines of symmetry 0 (usually)
  • Isosceles trapezium; Sides 1 pair parallel; non-parallel sides equal; Angles 2 pairs equal; Diagonals equal; Lines of symmetry 1

Key fact: A square is a special rectangle and a special rhombus. Rectangles and rhombuses are special parallelograms.

Angles in a quadrilateral

A diagonal splits any quadrilateral into two triangles. Each triangle has angles adding up to 180°, so the four angles of a quadrilateral add up to 2 × 180° = 360°.

Worked example: the missing angle

A quadrilateral has angles 85°, 110°, 72° and x.
85 + 110 + 72 = 267. x = 360° − 267° = 93°

Answer: x = 93°

Worked example: a parallelogram

One angle of a parallelogram is 64°. Find the other three angles.
Opposite angles are equal: 64°. The other two are equal and 64 + 64 + 2a = 360, so 2a = 232, a = 116°.

Answer: 64°, 116°, 116°

Worked example: a kite

A kite has angles 120° and 50° on its line of symmetry. The other two angles are equal. Find them.
120 + 50 = 170. 360 − 170 = 190. 190 ÷ 2 = 95

Answer: 95° each

Using algebra

Worked example: angles in terms of x

The angles of a quadrilateral are x, 2x, 3x and 4x. Find the largest angle.
x + 2x + 3x + 4x = 360, so 10x = 360 and x = 36.
Largest angle = 4 × 36 = 144°

Answer: 144°

Key fact: In a parallelogram (and a rhombus) the two angles next to each other add up to 180°.

Common mistakes

  • Using 180° for a quadrilateral. The angles in a quadrilateral add up to 360°.
  • Thinking a rhombus has right angles. Only the square has 4 equal sides and 4 right angles.
  • Mixing up the equal angles of a kite: they are the pair between the unequal sides, not on the line of symmetry.
  • Stopping at x when the question asks for an angle such as 4x.

Revision questions

Try these, then check your answers below each one.

Question 1. Two angles lie on a straight line. One of them is 117°. Work out the other angle.

Answer: 63 °

Working: 180 - 117 = 63°

Question 2. Four angles meet at a point. Three of them are 95°, 130° and 70°. Work out the fourth angle.

Answer: 65 °

Working: 360 - (95 + 130 + 70) = 360 - 295 = 65°

Question 3. Which quadrilateral has exactly one pair of parallel sides? (Choose from: Trapezium, Parallelogram, Kite, Rhombus.)

Answer: Trapezium

Working: A trapezium has exactly one pair of parallel sides.

Related lessons

Continue in the full course

This lesson is section 6.1 (EGCMF0601) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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