The angles of a quadrilateral add up to 360°. Therefore, if the opposite angles are equal, what can we say about adjacent angles? Is the converse of your answer true? Conclude that the result “if the opposite angles of a quadrilateral are equal, then it is a parallelogram” can also be stated as follows. “If each pair of adjacent angles in a quadrilateral ABCD …then ABCD is a parallelogram.” Fill in the blank.
Step-by-step solution
Idea: Use ∠A + ∠B + ∠C + ∠D = 360° both ways: equal opposite angles force adjacent sums of 180°, and adjacent sums of 180° force equal opposite angles.
- If ∠A = ∠C and ∠B = ∠D: 360° = 2(∠A + ∠B), so ∠A + ∠B = 180°. In the same way ∠B + ∠C = ∠C + ∠D = ∠D + ∠A = 180°. Each pair of adjacent angles is supplementary.1 mark
- Converse: if ∠A + ∠B = 180° and ∠B + ∠C = 180°, then ∠A = 180° − ∠B = ∠C. If also ∠C + ∠D = 180°, then ∠B = 180° − ∠C = ∠D. So the opposite angles are equal: the converse is true.1 mark
- So “opposite angles equal” and “each pair of adjacent angles adds to 180°” say the same thing. The result reads: if each pair of adjacent angles in a quadrilateral ABCD adds up to 180°, then ABCD is a parallelogram.1 mark
Answer to write in the exam
∠A = ∠C, ∠B = ∠D ⇒ 2(∠A + ∠B) = 360° ⇒ ∠A + ∠B = 180°; similarly every adjacent pair = 180°
Converse: ∠A + ∠B = 180° = ∠B + ∠C ⇒ ∠A = ∠C; ∠B + ∠C = 180° = ∠C + ∠D ⇒ ∠B = ∠D (true)
∴ “If each pair of adjacent angles in a quadrilateral ABCD is supplementary (adds up to 180°), then ABCD is a parallelogram.”
Common mistakes that cost marks
- Writing “adjacent angles are equal”. That is true only for a rectangle.
- Checking only one adjacent pair. One pair adding to 180° gives just one pair of parallel sides (a trapezium); each pair is needed.
- Saying the converse is false without testing it; the short algebra above shows it is true.
How this can come in the exam
In quadrilateral ABCD, ∠A + ∠B = 180° and ∠B + ∠C = 180°. Which must be true?
- ∠A = ∠B
- ∠A = ∠C
- ∠A + ∠C = 180°
- ∠B = ∠C
Show answer
(B) ∠A = ∠C
∠A = 180° − ∠B = ∠C.
Try one yourself
The angles of a quadrilateral, in order, are 75°, 105°, 75°, 105°. Show that each pair of adjacent angles is supplementary and name the quadrilateral.
Show answer
75 + 105 = 180 for every adjacent pair, so the quadrilateral is a parallelogram (opposite angles are also equal).
More questions like this
- A quadrilateral whose diagonals bisect each other is a parallelogram.
- A quadrilateral with one pair of equal and parallel opposite sides is a parallelogram.
- Suppose in a quadrilateral ABCD we have AB ‖ DC and AB = DC. Let E be the intersection point of the diagonals AC and BD. (Why must the diagonals intersect?)
- True or false?
- The diagonal AC of a parallelogram ABCD bisects ∠A. Show that it also bisects ∠C and that ABCD is a rhombus.
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