One definition, and every other property follows from it.
The problem
In parallelogram ABCD, side AB = 3x − 4 and CD = x + 8. Find the length of AB.
Opposite sides of a parallelogram are equal: 3x − 4 = x + 8 gives 2x = 12, so x = 6 and AB = 3(6) − 4 = 14. B is x itself, not the length. Check with CD: 6 + 8 = 14. ✓
| Property | Detail |
|---|---|
| opposite sides | parallel and equal |
| opposite angles | equal |
| consecutive angles | add to 180° |
| diagonals | bisect each other |
| all four angles | add to 360° |
The consecutive-angle rule is just co-interior angles between two parallel sides — nothing new to learn.
∠A = 65°
given
∠C = 65°
opposite to A
∠B = 180 − 65 = 115°
consecutive to A
∠D = 115°
opposite to B
Two sizes only, exactly as with parallel lines cut by a transversal.
∠A and ∠B share a side, so they are supplementary, not equal. Equal angles sit across the shape from each other.
AB = CD
opposite sides
3x − 4 = x + 8
the equation
2x = 12, x = 6
solve
AB = 3(6) − 4 = 14
substitute back
Check with the other expression: 6 + 8 = 14 ✓
Adjacent sides in the ratio 2 : 3, perimeter 50. Opposite sides are equal, so the perimeter holds two of each: 2(2k + 3k) = 50, 10k = 50, k = 5. The sides are 10 and 15 — not 20 and 30, which forgets the “two of each”.
each diagonal is cut in half
the two diagonals being equal
that needs a rectangle
diagonals meet at M
the crossing point
AM = MC and BM = MD
each is halved
but AC ≠ BD in general
a common wrong assumption
| Shape | Adds | Diagonals |
|---|---|---|
| parallelogram | — | bisect each other |
| rectangle | four right angles | also equal |
| rhombus | four equal sides | also perpendicular |
| square | both of the above | equal and perpendicular |
Every rectangle, rhombus and square is a parallelogram — so all four inherit its properties.
“Every square is a rhombus” is true. “Every rhombus is a square” is false. Exam questions on this family are usually testing exactly that direction.
A rhombus diagonal cuts its corner angle in half, and the two diagonals cross at 90°. A rectangle’s diagonals are equal, so their halves are equal too.
Rhombus ABCD with ∠B = 70°. Find ∠DAC.
∠A = 180 − 70 = 110°
consecutive angles
∠DAC = 110 ÷ 2 = 55°
diagonal AC bisects ∠A
The same rhombus idea with lengths: diagonals 12 and 16. Find the perimeter.
halves: 6 and 8
diagonals bisect each other
side = √36 + 64 = 10
they meet at a right angle
perimeter = 4 × 10 = 40
four equal sides
Rectangle: the diagonals meet at O and OA = OB = OC = OD, so all four small triangles are isosceles. If ∠AOB = 100°, then ∠OAB = ∠OBA = (180 − 100) ÷ 2 = 40°.
In parallelogram ABCD, ∠A = (3x + 8)° and ∠B = (5x − 4)°. Find both.
(3x + 8) + (5x − 4) = 180
A and B are consecutive
8x + 4 = 180, x = 22
− 4, ÷ 8
∠A = 74°, ∠B = 106°
74 + 106 = 180 ✓
Different from step 3: neighbouring letters share a side, so the equation is “total 180”, not “equal”.
The diagonals of parallelogram ABCD meet at M, with AM = 4y − 9 and MC = 2y + 5. Find AC.
4y − 9 = 2y + 5
M is the midpoint of AC
2y = 14, y = 7
− 2y, + 9, ÷ 2
AM = 19, AC = 2 × 19 = 38
double the half
Different: the expressions are halves. The answer is the whole diagonal, so one more line doubles it.
In rectangle ABCD, AC = 3z − 2 and BD = z + 14. Find AC.
3z − 2 = z + 14
a rectangle’s diagonals are equal
2z = 16, z = 8
solve
AC = 3(8) − 2 = 22
BD = 8 + 14 = 22 ✓
In a plain parallelogram the two diagonals are usually different lengths, so AC = BD is not an equation we may write. Only the two halves of the same diagonal are equal.
An angle bisector makes an isosceles triangle. Parallelogram ABCD has AD = 7 and AB = 12; the bisector of ∠A meets DC at E. Find EC.
∠DEA = ∠EAB
alternate angles, DC ∥ AB
∠EAB = ∠DAE
AE bisects ∠A
so DE = AD = 7
△ADE has two equal angles
EC = 12 − 7 = 5
DC = AB = 12
The fourth vertex from coordinates. A(2, 1), B(7, 2), C(9, 6) are three vertices of parallelogram ABCD. Find D.
A + C = B + D
the diagonals share a midpoint
D = (2 + 9 − 7, 1 + 6 − 2)
D = A + C − B
D = (4, 5)
check: AB and DC both move (5, 1) ✓
ABCD means A and C are opposite corners, so D is the vertex opposite B. Adding the wrong pair (say A + B − C) gives a real parallelogram, just not ABCD — and it will be among the choices.
Expect angles or sides as expressions (opposite equal, consecutive 180), a diagonal split at the crossing point, or a rhombus or rectangle where the diagonals do extra work. Ask which shape it is before you use a diagonal fact — “equal” needs a rectangle, “perpendicular” and “bisects the angle” need a rhombus. Ten-second check: the four angles you end with total 360°.
Find x
In parallelogram PQRS, ∠P = (2x + 20)° and ∠Q = (3x − 10)°. Find x.
P and Q are consecutive angles, so they are supplementary: (2x + 20) + (3x − 10) = 180 gives 5x = 170 and x = 34. C sets them equal — but equality holds for opposite angles, and these two are adjacent.
