Every special quadrilateral is a parallelogram with something extra.
Find the angle
In parallelogram ABCD, ∠A = 72°. What is ∠B?
In a parallelogram, opposite angles are equal and consecutive angles are supplementary. A and B are next to each other, so ∠B = 180 − 72 = 108°. A is ∠C, the angle opposite A. Which rule applies depends entirely on whether the two angles are adjacent or opposite — check the letters.
| Property | Meaning |
|---|---|
| Opposite sides parallel | the definition |
| Opposite sides equal | AB = DC and AD = BC |
| Opposite angles equal | ∠A = ∠C and ∠B = ∠D |
| Consecutive angles supplementary | ∠A + ∠B = 180° |
| Diagonals bisect each other | they cut each other in half |
The diagonals of a parallelogram cut each other in half, but they are not the same length as each other. Equal diagonals is a rectangle's property, which is stronger.
| Shape | Adds | So its diagonals |
|---|---|---|
| Rectangle | four right angles | are equal |
| Rhombus | four equal sides | are perpendicular, and bisect the angles |
| Square | both of the above | equal AND perpendicular |
A square is a rectangle and a rhombus at once, so it inherits every property of both. That is why it has the longest list and the fewest surprises.
AB = 3x + 4, DC = 5x − 10
opposite sides of a parallelogram
3x + 4 = 5x − 10
equal
x = 7, so AB = 25
and DC = 25 ✓
∠A = (2y + 10)°, ∠B = (3y)°
consecutive
2y + 10 + 3y = 180
supplementary
y = 34, angles 78° and 102°
adding to 180 ✓
Perimeter uses the same fact: opposite sides are equal, so P = 2(AB + AD). Sides of 7 cm and 4 cm give 2(7 + 4) = 22 cm.
Any one of these five is enough.
A trapezoid has one pair of parallel sides and is not a parallelogram. That pair must be both parallel and equal for the last test to work.
Reverse any true statement here and it becomes false. The special shapes are examples of the general one, never the other way round.
A rhombus's diagonals bisect each other at right angles, so they cut it into four right triangles whose legs are HALF the diagonals.
A rhombus has diagonals of 10 cm and 24 cm. Find its side and perimeter.
legs 5 cm and 12 cm
half of each diagonal
side² = 5² + 12² = 169
Pythagoras in one triangle
side = 13 cm
√169
perimeter = 4 × 13 = 52 cm
four equal sides
10² + 24² = 676, and √676 = 26 cm — twice the real side. The right triangle's legs run from a corner only to the centre, so each is half a diagonal.
Each one uses a property from the tables, written as an equation.
Diagonals of parallelogram ABCD meet at E. AE = 3x − 2 and EC = x + 8. Find AC.
3x − 2 = x + 8
the diagonals bisect each other
x = 5, so AE = 13
check: 5 + 8 = 13 ✓
AC = 2 × 13 = 26
AE is only half
AE and EC are halves of the same diagonal; the whole diagonal is their sum.
In rectangle PQRS, PR = 5y − 6 and QS = 3y + 8. Find PR.
5y − 6 = 3y + 8
a rectangle's diagonals are equal
2y = 14, so y = 7
solve
PR = 5(7) − 6 = 29
check: 3(7) + 8 = 29 ✓
This equation is only allowed because it is a rectangle; in a general parallelogram the diagonals differ.
Two consecutive angles of a parallelogram are in the ratio 2 : 3. Find them.
2k + 3k = 180
consecutive: supplementary
k = 36
5k = 180
72° and 108°
check: 72 + 108 = 180 ✓
A ratio needs a multiplier k. The four angles are 72°, 108°, 72°, 108°.
Expect an angle from a consecutive or opposite angle, a side or diagonal written in x, a "what must it be" item from diagonal properties, and a rhombus side from its diagonals. The fastest route is to match the property to the shape first — opposite sides equal, consecutive angles supplementary, diagonals bisect — then write one equation. The 10-second check: adjacent angles add to 180, opposite ones match, and a half-diagonal doubled gives the whole.
Identify the shape
A parallelogram has diagonals that are equal in length and perpendicular to each other. What must it be?
Equal diagonals make it a rectangle; perpendicular diagonals make it a rhombus. A shape that is both is a square. A and B each use only one of the two clues — the question gives both, and both must hold. C is impossible, since a trapezoid is not a parallelogram at all.
Hint: each diagonal property identifies one special shape. Two properties means the shape must satisfy both at once.
