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Parallelogram · finding an unknown angle

P6 · Geometry · PSLE Maths

Question

Mei Ling drew a parallelogram ABCD in art class and drew the diagonal AC. Given that ∠ABC = 70° and ∠BAC = 45°, find the size of ∠ACD.

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Hint

First use adjacent angles in a parallelogram to find ∠DAB, then see how diagonal AC splits this angle.

Show the answer and step-by-step explanation

Answer: ∠ACD 45 °

  1. Draw figure — First draw the figure clearly: ABCD is a parallelogram and AC is a diagonal. We know ∠ABC = 70° and ∠BAC = 45°. We need to find ∠ACD.
  2. Adjacent angles — In a parallelogram, adjacent angles add up to 180°. So ∠DAB and ∠ABC add up to 180°.
  3. Split angle — The diagonal AC splits ∠DAB into two parts. ∠BAC is 45°, so the remaining ∠DAC is 110° − 45° = 65°.
  4. Opposite angles — Opposite angles in a parallelogram are equal. ∠CDA and ∠ABC are opposite angles, so ∠CDA is also 70°.
  5. Triangle sum — Now look at triangle ADC: its three interior angles add up to 180°. So x = 180° − 65° − 70° = 45°. 📌 Key idea: in a parallelogram, use adjacent angles and opposite angles first, then use the angle sum of a triangle.

Method: In a parallelogram, adjacent angles add up to 180° and opposite angles are equal. Find the angle split by the diagonal, then use the angle sum of a triangle.

Practice: 3 similar questions

  1. Ali drew a parallelogram ABCD and drew the diagonal AC. Given that ∠ABC = 64° and ∠BAC = 38°, find ∠ACD.
    Answer

    38 ° — Adjacent angles are supplementary: ∠DAB = 180° − 64° = 116°. The diagonal splits the angle: ∠DAC = 116° − 38° = 78°. Opposite angles are equal: ∠CDA = ∠ABC = 64°. In triangle ADC, ∠ACD = 180° − 78° − 64° = 38°.

  2. Siti drew a rhombus WXYZ and drew the diagonal WY. Given that ∠WXY = 76°, find ∠WYZ.
    Answer

    52 ° — A rhombus is also a parallelogram, so adjacent angles are supplementary: ∠XYZ = 180° − 76° = 104°. In a rhombus, diagonal WY bisects ∠XYZ. So ∠WYZ = 104° ÷ 2 = 52°.

  3. Raju drew a trapezium ABCD with AB ∥ CD and drew the diagonal AC. Given that ∠BCD = 86° and ∠BAC = 34°, find ∠ACB.
    Answer

    52 ° — Since AB ∥ CD, ∠ABC + ∠BCD = 180°. ∠ABC = 180° − 86° = 94°. In triangle ABC, ∠ACB = 180° − 94° − 34° = 52°.

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