Quadrilaterals · Ages 14–15
Properties of a parallelogram
Opposite sides and angles equal, diagonals bisect each other, next-door angles make 180°.
How to do it
Method
Opposite sides and angles equal, diagonals bisect each other, next-door angles make 180°
∠A = 70°: ∠C = 70°, ∠B = 180° − 70° = 110°
Another way
Two congruent triangles
△ABC ≅ △CDA (ASA), AB = CD, ∠B = ∠D
In full
A parallelogram is a quadrilateral with both pairs of opposite sides parallel. It has four properties: opposite sides equal (opp. sides of // gram); opposite angles equal (opp. ∠s of // gram); diagonals bisecting each other (diags. of // gram), so they cross at their middles; and next-door angles adding to 180° (int. ∠s, AB // DC). ∠A = 70° gives ∠C = 70° (opposite) and ∠B = ∠D = 180° − 70° = 110° (interior angles). With the diagonals AC and BD crossing at E and AE = 4 cm: EC = 4 cm and AC = 8 cm. Another way: a diagonal cuts it into two congruent triangles (ASA), whose matching sides and angles give the same facts. Careful: the diagonals bisect each other but needn't be equal or at right angles.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
ABCD is a parallelogram. x =
Show the answer and working
Answer: 75°
∠C and ∠A are opposite: x = 75° (opp. ∠s of // gram).
Example 2
ABCD is a parallelogram whose diagonals cross at E, with AE = 3 cm and BE = 4 cm. x = , BD = cm
Fill in each blank
Show the answer and working
Answer: 3, 8
The diagonals bisect each other (diags. of // gram): x = AE = 3, BD = 2 × 4 = 8 cm.
Example 3
Which of these is not always a property of a parallelogram?
- A. opposite angles are equal
- B. the diagonals are equal
- C. the diagonals bisect each other
- D. opposite sides are equal
Choose the right answer
Show the answer and working
Answer: B. the diagonals are equal
Equal diagonals belong to rectangles; a general parallelogram's diagonals differ in length.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.ABCD is a parallelogram. x =
Answer:
2.ABCD is a parallelogram whose diagonals cross at E, with AE = 8 cm and BE = 4 cm. x = , BD = cm
Fill in each blank
Answer:
3.In the parallelogram ABCD, ∠A = (4x − 9)° and ∠C = (5x − 26)°. x =
Answer:
4.The parallelogram ABCD has a perimeter of 52 cm, and AB is 4 cm longer than BC. AB =
Answer:
5.Which of these is not always a property of a parallelogram?
- A. the diagonals bisect each other
- B. the diagonals are equal
- C. opposite angles are equal
- D. opposite sides are equal
Choose the right answer
Answer:
All the answers
- 1. 60°
- 2. 8, 8
- 3. 17
- 4. 15 cm
- 5. B. the diagonals are equal
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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