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Polygons

Polygons

Angle facts and reasons

Formulas Core + Extended 7 min read

On this page
  1. In this note
  2. Formula bank
  3. The angle facts
  4. Ways they test it
  5. Mistakes to avoid
  6. Summary
  7. Practice
  8. Can I do this?
  9. Quiz

In this note

Syllabus 4.1 and 4.6: angle facts
  • 4.6Use angles on a straight line, angles at a point, vertically opposite angles and the angle sums of triangles and quadrilaterals.
  • 4.1Use isosceles and equilateral triangles and three-letter notation (angle ABC).
  • 4.6Give a correct geometrical reason for every step.

Next: Angles in parallel lines.

Formula bank

Straight lineLearn itangles on a straight line add up to 180°Use it when angles meeting on a line.
PointLearn itangles at a point add up to 360°Use it when angles all the way round.
Vertically oppositeLearn itvertically opposite angles are equalUse it when two straight lines crossing.
Triangle / quadrilateralLearn itangles in a triangle add up to 180°; in a quadrilateral, 360°Use it when angles inside a shape.
IsoscelesLearn itbase angles of an isosceles triangle are equalthe angles opposite the equal sides.Use it when two equal sides (ticks).
Exterior angleLearn itexterior angle of a triangle = sum of the two interior opposite anglesUse it when a triangle side extended.

The angle facts

Angle ABC is the angle at B (the middle letter), between BA and BC. Diagrams marked NOT TO SCALE must be worked out, never measured.

Angles on a straight line47°x°68°
47 + x + 68 = 180, so x = 65.
Vertically opposite anglesaabb
Opposite angles are equal; a + b = 180°.
Exterior angle of a triangle40°70°70°110°ABCD
AB = CB (ticks), so the base angles at A and C are equal.

Angle ABC = 40°, so each base angle is \((180-40)\div2=70^\circ\). Angle BCD = 180° − 70° = 110°, which also equals 40° + 70° (the exterior angle).

General methodHow to find an unknown angle (with reasons)

  1. Mark every angle you know on the diagram; look for equal sides (isosceles) and straight lines.
  2. Find the angle that has only one unknown next to it, using one fact.
  3. Write the value AND the reason in full words (from the formula bank).
  4. Repeat until you reach the angle asked for. With algebra, form one equation and solve.

Example The angles of a quadrilateral are \(x^\circ\), \((x+20)^\circ\), \(2x^\circ\) and \((2x-20)^\circ\). Find the largest angle.

Add them and set equal to 360°.

  1. \(6x=360\)
  2. \(x=60\)
  3. angles 60°, 80°, 120°, 100°

Answer 120°

Try it yourself

1
Two angles of a triangle are 35° and 90°. The third?
Show answer

Answer 55

2
Three angles of a quadrilateral are 90°, 125°, 85°. The fourth?
Show answer

Answer 60

3
An isosceles triangle has apex angle 36°. Each base angle?
Show answer

Answer 72

Ways they test it

Type 1 Find x with reasons

"Find the value of x. Give a geometrical reason for each answer." (2-4 marks)

Spot it 'giving a reason'.

  1. Value, then the fact in full words.
  2. One reason per step.

For example two lines cross; one angle 52°

  1. opposite angle 52° (vertically opposite angles are equal)
  2. neighbour 128° (angles on a straight line)

Answer 52° and 128°

Type 2 Angles at a point with algebra

"Angles 110°, \(2x^\circ\), \((x+40)^\circ\) meet at a point. Find x." (2 marks)

Spot it algebra around a point.

  1. Sum = 360.
  2. Solve.

For example \(3x+150=360\)

    Answer x = 70

    Type 3 Isosceles + exterior

    "AB = CB, angle ABC = 40°. Find angle BCD." (3 marks)

    Spot it ticks on two sides.

    1. Base angles first.
    2. Then the straight line (or exterior angle).

    Mistakes to avoid

    • ✗ Measuring the diagram✓ calculateDiagrams are NOT TO SCALE.
    • ✗ 'opposite angles' as a reason✓ 'vertically opposite angles are equal'The full wording earns the mark.
    • ✗ Equal angles at the wrong corners✓ base angles are opposite the equal sidesCheck the ticks.
    • ✗ Giving x when the angle is asked✓ substitute backRead the question.

    Summary

    • Line 180°, point 360°, vertically opposite equal.
    • Triangle 180°, quadrilateral 360°.
    • Isosceles: base angles equal; equilateral: 60° each.
    • Exterior angle = sum of the two interior opposite angles.

    If you see …

    • "give a reason"write the fact in full words.
    • tick marks on two sidesisosceles: equal base angles.

    Practice

    Practice: easy, medium, hard

    Easy

    1
    Angles on a straight line: 47° and 68°. The third angle x?
    Show answer

    Answer 65

    2
    Two straight lines cross; one angle is 52°. The angle next to it?
    Show answer

    Answer 128

    3
    Angles 110°, 2x° and (x + 40)° meet at a point. Find x.
    Show answer

    Answer 70

    Medium

    1
    AB = CB and angle ABC = 40°. Find angle BCA.
    Guide me

    What is it asking? an angle

    What do we know? isosceles: A and C equal

    Plan (180 − 40) ÷ 2

    Show working
    1. 70°

    Answer 70

    2
    AC is extended to D. Find angle BCD in the same triangle.
    Guide me

    What is it asking? an angle

    What do we know? straight line at C

    Plan 180 − 70

    Show working
    1. 110°

    Answer 110

    3
    A quadrilateral has angles x°, (x + 20)°, 2x°, (2x − 20)°. Find x.
    Guide me

    What is it asking? x

    What do we know? sum 360

    Plan 6x = 360

    Show working
    1. \(6x=360\)

    Answer 60

    Hard

    1
    PQR is a straight line. QRS is a triangle with QS = RS and angle QSR = 50°. Find angle PQS.
    Guide me

    What is it asking? an angle

    What do we know? base angles (180 − 50) ÷ 2 = 65

    Plan straight line at Q

    Show working
    1. angle SQR = 65°
    2. angle PQS = 180 − 65 = 115°

    Answer 115

    PartAnswer / workingMarksGuidance
    65B1
    180 − 65M1
    115A1
    Total: 3 marks
    2
    The angles of a triangle are x°, 2x° and (3x − 30)°. Find the largest angle.
    Guide me

    What is it asking? an angle

    What do we know? 6x − 30 = 180

    Plan x = 35

    Show working
    1. \(6x=210\Rightarrow x=35\)
    2. angles 35°, 70°, 75°

    Answer 75

    PartAnswer / workingMarksGuidance
    6x − 30 = 180M1
    x = 35A1
    75A1
    Total: 3 marks
    3
    An isosceles triangle has base angles of 2x° and apex angle x°. Find x.
    Guide me

    What is it asking? x

    What do we know? 2x + 2x + x = 180

    Plan 5x = 180

    Show working
    1. \(5x=180\)
    2. \(x=36\)

    Answer 36

    PartAnswer / workingMarksGuidance
    5x = 180M1
    36A1
    Total: 2 marks

    Can I do this?

    Anything unticked: revise that section first. All ticked: take the quiz.

    Quiz

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