Angle facts Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Angle facts questions. See exactly how to solve problems on angles on a straight line, reasoning, three angles, angles at a point.

angles on a straight linereasoningthree anglesangles at a pointforming an equationvertically opposite angles
GCSE Foundation70 questionsStep-by-step solutions
Question 1
1 markeasy
AOBAOB is a straight line and CC is a point above the line. Angle AOC=130∘AOC = 130^\circ. Work out the size of angle COBCOB.

Worked solution

  1. Use the angle fact for a straight line

    ∠AOC+∠COB=180∘\angle AOC + \angle COB = 180^\circ

    Angles on a straight line add up to 180∘180^{\circ}.

  2. Subtract the known angle

    ∠COB=180∘−130∘=50∘\angle COB = 180^\circ - 130^\circ = 50^\circ

    Taking 130∘130^{\circ} away from 180∘180^{\circ} leaves the angle you want.

  3. State the answer

    ∠COB=50∘\angle COB = 50^\circ

    Angle COBCOB measures 50∘50^{\circ}.

Answer
50∘50^\circ
Question 2
2 markseasy
POQPOQ is a straight line. The rays OROR and OSOS are drawn above the line so that angle POR=45∘POR = 45^\circ and angle ROS=85∘ROS = 85^\circ. Work out the size of angle SOQSOQ.

Worked solution

  1. Use the angle fact for a straight line

    45∘+85∘+∠SOQ=180∘45^\circ + 85^\circ + \angle SOQ = 180^\circ

    Angles on a straight line add up to 180∘180^{\circ}.

  2. Subtract the two known angles

    ∠SOQ=180∘−45∘−85∘=50∘\angle SOQ = 180^\circ - 45^\circ - 85^\circ = 50^\circ

    The three angles fill the straight line, so take both known angles from 180∘180^{\circ}.

  3. State the answer

    ∠SOQ=50∘\angle SOQ = 50^\circ

    Angle SOQSOQ measures 50∘50^{\circ}.

Answer
50∘50^\circ
Question 3
2 markseasy
Three rays OAOA, OBOB and OCOC are drawn from the point OO. Angle AOB=150∘AOB = 150^\circ and angle BOC=120∘BOC = 120^\circ. The three angles fill the whole turn at OO. Work out the size of angle AOCAOC.

Worked solution

  1. Use the angle fact for a point

    150∘+120∘+∠AOC=360∘150^\circ + 120^\circ + \angle AOC = 360^\circ

    Angles at a point add up to 360∘360^{\circ}.

  2. Subtract the two known angles

    ∠AOC=360∘−150∘−120∘=90∘\angle AOC = 360^\circ - 150^\circ - 120^\circ = 90^\circ

    A full turn is 360∘360^{\circ}, so take the two known angles away from 360∘360^{\circ}.

  3. State the answer

    ∠AOC=90∘\angle AOC = 90^\circ

    Angle AOCAOC measures 90∘90^{\circ}, a right angle.

Answer
90∘90^\circ
Question 4
2 markseasy
Four angles meet at a point and fill the whole turn. Three of them are 90∘90^\circ, 115∘115^\circ and 65∘65^\circ. The fourth angle is x∘x^\circ. Work out the value of xx.

Worked solution

  1. Use the angle fact for a point

    90∘+115∘+65∘+x∘=360∘90^\circ + 115^\circ + 65^\circ + x^\circ = 360^\circ

    Angles at a point add up to 360∘360^{\circ}.

  2. Subtract the three known angles

    x=360−90−115−65=90x = 360 - 90 - 115 - 65 = 90

    The three known angles total 270∘270^{\circ}, and 360360 minus 270270 is 9090.

  3. State the answer

    x=90x = 90

    The fourth angle is 90∘90^{\circ}.

Answer
x=90x = 90
Question 5
1 markeasy
Two straight lines ABAB and CDCD cross at the point OO. Angle AOC=72∘AOC = 72^\circ. Work out the size of angle BODBOD.

Worked solution

  1. Identify the vertically opposite pair

    ∠AOC and ∠BOD\angle AOC \text{ and } \angle BOD

    Angle AOCAOC and angle BODBOD are on opposite sides of the crossing point, so they are vertically opposite.

  2. Use the angle fact

    ∠BOD=∠AOC=72∘\angle BOD = \angle AOC = 72^\circ

    Vertically opposite angles are equal.

  3. State the answer

    ∠BOD=72∘\angle BOD = 72^\circ

    Angle BODBOD measures 72∘72^{\circ}.

Answer
72∘72^\circ

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