A window, a book cover and a floor tile are all quadrilaterals, shapes enclosed by four sides, and a quadrilateral whose two pairs of opposite sides are both parallel is a parallelogram.
In a parallelogram, opposite sides are equal in length, opposite angles are equal in size, and the two diagonals cut each other in half.
Add the condition that all four angles are right angles and it becomes a rectangle; add the condition that all four sides are equal and it becomes a rhombus; with both conditions it becomes a square.
In the picture, raising ∠A to 90° or setting the side ratio to 1 lets you watch the same parallelogram change its name and properties each time it gains one condition.
From parallelogram to square
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Quadrilateral ABCD turning into a parallelogram, rectangle, rhombus or square as ∠A and the side ratio change
65°
0.7
👀 How the quadrilaterals are related
①A parallelogram is a quadrilateral whose two pairs of opposite sides are parallel.
②When one angle of a parallelogram becomes 90°, all four angles are 90° and it is a rectangle; its diagonals become equal in length.
③When two adjacent sides of a parallelogram become equal, all four sides are equal and it is a rhombus; its diagonals become perpendicular.
④With both conditions it is a square, which has the properties of both the rectangle and the rhombus.
Properties of a parallelogram
Three properties of a parallelogram
opposite sides equal · opposite angles equal · diagonals bisect each other
All three follow from the fact that a diagonal splits the parallelogram into two congruent triangles (ASA).
💡 Conditions for a parallelogram
①Both pairs of opposite sides are parallel (the definition).
②Both pairs of opposite sides are equal in length.
③Both pairs of opposite angles are equal.
④The diagonals bisect each other.
⑤One pair of opposite sides is parallel and equal in length. Checking any one of these five is enough.
Rectangle, rhombus and square
Extra properties of a rectangle
all four angles are 90°, and the diagonals are equal in length
A parallelogram with one angle of 90° is a rectangle.
Extra properties of a rhombus
all four sides are equal, and the diagonals are perpendicular
A parallelogram with two equal adjacent sides is a rhombus.
Square
a quadrilateral that is both a rectangle and a rhombus = square
With four equal sides and four 90° angles, its diagonals are equal and bisect each other at right angles.
How the quadrilaterals contain one another
📝 Inclusion relations
①square ⊂ rectangle ⊂ parallelogram.
②square ⊂ rhombus ⊂ parallelogram.
③A square is both a rectangle and a rhombus.
④Whether a parallelogram counts as a trapezoid depends on the definition: with "at least one pair of parallel sides" it does, with "exactly one pair" it does not.
Worked examples
Example 1
In a parallelogram one interior angle is 110°. Find the adjacent interior angle.
1
Adjacent interior angles of a parallelogram sum to 180°.
adjacent angle = 180° − 110°
2
Compute.
= 70°
▸ 70°
In a parallelogram opposite angles are equal and adjacent ones sum to 180°.
Example 2
In parallelogram ABCD, AB = 8 and BC = 5. Find the perimeter.
1
A parallelogram has two pairs of equal opposite sides.
perimeter = 2 × (8 + 5)
2
Compute.
= 26
▸ 26
Opposite sides of a parallelogram are equal in length.
Exam checklist
How the properties build up
parallelogram → rectangle or rhombus → square
At each step, be precise about which single condition is added.
Grade-8 school exam type
In a parallelogram, adjacent angles ∠A and ∠B are in ratio 2 : 3. What is ∠A?
①60°
②72°
③90°
④108°
⑤120°
▸ ② 72°
1
Adjacent angles sum to 180°, so 2k + 3k = 180°.
2k + 3k = 180° ⇒ 5k = 180°
2
k = 36°, so ∠A = 2k.
∠A = 2 × 36° = 72°
🎯 Exam Points
①To prove a quadrilateral is a parallelogram, showing any one of the five conditions is enough.
②A rectangle is a parallelogram with one 90° angle, or equivalently with equal diagonals.
③A rhombus is a parallelogram with two equal adjacent sides, or equivalently with perpendicular diagonals.
④Area: parallelogram = base × height; rhombus = (product of the diagonals) ÷ 2.
⑤Two adjacent angles of a parallelogram add up to 180°.