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Angles · straight line and vertically opposite angles

P6 · Geometry · PSLE Maths

Question

Mei Ling drew a design in her notebook. Two straight lines AB and CD intersect at O. Ray OE is inside ∠AOC. Given that ∠AOD = 126° and ∠COE = 20°, find ∠AOE, marked x in the figure.

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Hint

First find the angle vertically opposite to 126°, then use angles on a straight line add up to 180°.

Show the answer and step-by-step explanation

Answer: x 34 °

  1. Read the figure — The two straight lines AB and CD intersect at O. Ray OE splits the lower-left angle ∠AOC into two parts: 20° and x.
  2. Opposite angles — ∠AOD and ∠BOC are vertically opposite angles, so they are equal. Thus ∠BOC is also 126°.
  3. Use straight line — A, O and B lie on a straight line, so ∠AOC and ∠COB add up to 180°.
  4. Find the whole angle — So the whole ∠AOC is: 180° − 126° = 54°.
  5. Subtract 20° — ∠AOC is split into x and 20°. So x = 54° − 20° = 34°. 📌 Key idea: use vertically opposite angles first, then angles on a straight line add up to 180°, and finally subtract the known small angle.

Method: When two straight lines intersect, vertically opposite angles are equal. Adjacent angles on a straight line add up to 180°. Find the whole ∠AOC first, then subtract the given 20°.

Practice: 3 similar questions

  1. Two straight lines AB and CD intersect at O. Ray OE is inside ∠AOC. Given that ∠AOD = 132° and ∠COE = 25°, find ∠AOE, marked x.
    Answer

    23 ° — ∠AOD and ∠BOC are vertically opposite angles, so ∠BOC = 132°. ∠AOC + ∠COB = 180°, so ∠AOC = 180° − 132° = 48°. x = 48° − 25° = 23°.

  2. Two straight lines PQ and RS intersect at T. Ray TU is inside ∠QTS. Given that ∠PTR = 118° and ∠STU = 31°, find ∠UTQ, marked x.
    Answer

    87 ° — ∠PTR and ∠QTS are vertically opposite angles, so ∠QTS = 118°. Ray TU splits ∠QTS into 31° and x. x = 118° − 31° = 87°.

  3. Two straight lines AB and CD intersect at O. Ray OE is inside ∠AOC. Given that ∠AOD = 120° and ∠COE is twice ∠AOE, find ∠AOE, marked x.
    Answer

    20 ° — ∠AOD and ∠BOC are vertically opposite angles, so ∠BOC = 120°. On straight line AB: ∠AOC = 180° − 120° = 60°. ∠AOC = x + 2x = 3x, so 3x = 60°. x = 60° ÷ 3 = 20°.

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