NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Properties of quadrilaterals

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A school in Perth is ordering custom quadrilateral-shaped desks for a new classroom. Each desk is designed as a parallelogram with opposite angles equal and adjacent angles supplementary.

One angle of the desk is given as 92 degrees, and the diagonal from this angle bisects it into two equal parts. What is the exact measure, in degrees, of one of the angles formed by the diagonal and an adjacent side?

Express your answer as an exact value.

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Squares belong to two broader families. A square is both a rectangle and a rhombus. The arrows mean “is a type of”, not a step in a calculation.RectangleRhombusSquare
A square is both a rectangle and a rhombus. The arrows mean “is a type of”, not a step in a calculation.

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Explanation

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Worked solution

  1. The desk is a parallelogram with one given angle of 92 degrees. In a parallelogram, opposite angles are equal and adjacent angles are supplementary (sum to 180°).
  2. Since adjacent angles are supplementary, the adjacent angle to 92° is 180° - 92° = 88°.
  3. The diagonal from the 92° angle bisects it into two equal parts. Therefore, each half is 92° ÷ 2 = 46°.
  4. The diagonal forms a triangle with the two adjacent sides of the parallelogram. In this triangle, one angle is 46° (from the bisected 92° angle), and another angle is 88° (the adjacent angle of the parallelogram).
  5. The sum of angles in any triangle is 180°. Therefore, the third angle in the triangle is 180° - 46° - 88° = 46°.
  6. The angle formed by the diagonal and an adjacent side is the angle between the diagonal and the side that forms the 88° angle of the parallelogram. This is the 46° angle found in the triangle.
  7. Thus, the measure of one of the angles formed by the diagonal and an adjacent side is 46 degrees.

Reference answer: 46 degrees

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