If the opposite angles of a quadrilateral are equal, then it is a parallelogram.
Step-by-step solution
To find: Prove that ABCD is a parallelogram
Idea: Use the angle sum 360° to show that adjacent angles add up to 180°. When co-interior angles on a transversal add to 180°, the two lines are parallel.
- Angles of a quadrilateral add to 360°: ∠A + ∠B + ∠C + ∠D = 360°. With ∠A = ∠C and ∠B = ∠D this is 2(∠A + ∠B) = 360°, and also 2(∠B + ∠C) = 360°.1 mark
- So ∠A + ∠B = 180° and ∠B + ∠C = 180°.½ mark
- AB is a transversal for BC and AD, and ∠A, ∠B are co-interior angles adding to 180°, so BC ‖ AD. BC is a transversal for AB and DC, and ∠B + ∠C = 180°, so AB ‖ DC. Hence ABCD is a parallelogram.1½ marks
Check: Example: angles 70°, 110°, 70°, 110° in order. Adjacent pairs: 70 + 110 = 180 ✓, so sides are parallel, as the proof says.
Answer to write in the exam
∠A + ∠B + ∠C + ∠D = 360° (angle sum of a quadrilateral)
∠A = ∠C, ∠B = ∠D ⇒ 2(∠A + ∠B) = 360° ⇒ ∠A + ∠B = 180°
Similarly ∠B + ∠C = 180°
∠A + ∠B = 180° ⇒ AD ‖ BC (co-interior angles, transversal AB)
∠B + ∠C = 180° ⇒ AB ‖ DC (co-interior angles, transversal BC)
∴ ABCD is a parallelogram.
Common mistakes that cost marks
- Writing ∠A + ∠C = 180°. Opposite angles are equal; it is adjacent angles that add to 180°.
- Naming the wrong transversal: for AD ‖ BC the transversal is AB (or CD), not AD itself.
- Assuming the sides are parallel first and then using parallel-line angle facts. That is circular.
How this can come in the exam
In quadrilateral ABCD, ∠A = ∠C = 65° and ∠B = ∠D. Then ∠B equals
- 65°
- 115°
- 125°
- 135°
Show answer
(B) 115°
2(65° + ∠B) = 360° ⇒ ∠B = 180° − 65° = 115°. (ABCD is then a parallelogram.)
Try one yourself
The angles of quadrilateral PQRS taken in order are (2a)°, (3a − 20)°, (2a)°, (3a − 20)°. Find a and show that PQRS is a parallelogram.
Show answer
2(2a + 3a − 20) = 360 ⇒ 5a = 200 ⇒ a = 40. Angles 80°, 100°, 80°, 100°: opposite angles equal, so PQRS is a parallelogram.
More questions like this
- 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.
- 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?
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