1
ΔABC ≅ ΔCDA
2
AB = CD and BC = AD
3
∠A = ∠C and ∠B = ∠D
4
AB ∥ CD and AB = CD ⇒ ABCD is a parallelogram
5
∠A = ∠C and ∠B = ∠D ⇒ ABCD is a parallelogram
6
AO = OC and BO = OD ⇒ ABCD is a parallelogram
FORMULA CHAPTER
Chapter 06 – Properties of a Parallelogram
WBBSEClass 09Mathematics6 formulas
Complete WBBSE Madhyamik Class 9 Properties of a Parallelogram theorem list for exam preparation. Learn equal opposite sides and angles, diagonal congruence, and converse tests using parallel sides, equal angles and bisecting diagonals. Review notation, common mistakes and clearly solved step-by-step examples for confident geometry revision and examinations.
View
A Diagonal Divides a Parallelogram into Congruent Triangles
ΔABC ≅ ΔCDA
SymbolMeaningUnit
AB, BC, CD, DASides of quadrilateral ABCDlength units
AC, BDDiagonals of quadrilateral ABCDlength units
∠A, ∠B, ∠C, ∠DInterior angles of ABCDdegrees
Worked example
Question: In parallelogram ABCD, prove that diagonal AC divides it into two congruent triangles.
Calculation:
Step 1: Opposite sides of a parallelogram are equal, so AB = CD and BC = AD.
Step 2: AC is common to △ABC and △CDA.
Step 3: The three corresponding sides are equal; therefore △ABC ≅ △CDA by SSS.
Answer: Diagonal AC divides ABCD into two congruent triangles.
Opposite Sides of a Parallelogram
AB = CD and BC = AD
SymbolMeaningUnit
AB, BC, CD, DASides of quadrilateral ABCDlength units
AC, BDDiagonals of quadrilateral ABCDlength units
∠A, ∠B, ∠C, ∠DInterior angles of ABCDdegrees
Worked example
Question: In parallelogram ABCD, AB = 8 cm and BC = 5 cm. Find CD and DA.
Calculation:
Step 1: AB = CD, so CD = 8 cm.
Step 2: BC = AD, so DA = 5 cm.
Answer: CD = 8 cm and DA = 5 cm.
Opposite Angles of a Parallelogram
∠A = ∠C and ∠B = ∠D
SymbolMeaningUnit
AB, BC, CD, DASides of quadrilateral ABCDlength units
AC, BDDiagonals of quadrilateral ABCDlength units
∠A, ∠B, ∠C, ∠DInterior angles of ABCDdegrees
Worked example
Question: In parallelogram ABCD, ∠A = 70°. Find ∠B, ∠C and ∠D.
Calculation:
Step 1: ∠C = ∠A = 70°.
Step 2: Adjacent angles are supplementary, so ∠B = 180° − 70° = 110°.
Step 3: ∠D = ∠B = 110°.
Answer: ∠B = 110°, ∠C = 70° and ∠D = 110°.
One Equal and Parallel Pair Test for a Parallelogram
AB ∥ CD and AB = CD ⇒ ABCD is a parallelogram
SymbolMeaningUnit
AB, BC, CD, DASides of quadrilateral ABCDlength units
AC, BDDiagonals of quadrilateral ABCDlength units
∠A, ∠B, ∠C, ∠DInterior angles of ABCDdegrees
Worked example
Question: In quadrilateral ABCD, AB ∥ CD and AB = CD = 6 cm. Identify the quadrilateral.
Calculation:
Step 1: AB and CD are opposite sides.
Step 2: They are parallel: AB ∥ CD.
Step 3: They are equal: AB = CD.
Answer: By the equal-and-parallel opposite-side test, ABCD is a parallelogram.
Opposite Angles Test for a Parallelogram
∠A = ∠C and ∠B = ∠D ⇒ ABCD is a parallelogram
SymbolMeaningUnit
AB, BC, CD, DASides of quadrilateral ABCDlength units
AC, BDDiagonals of quadrilateral ABCDlength units
∠A, ∠B, ∠C, ∠DInterior angles of ABCDdegrees
Worked example
Question: A quadrilateral ABCD has ∠A = ∠C = 70° and ∠B = ∠D = 110°. Prove it is a parallelogram.
Calculation:
Step 1: The first pair of opposite angles is equal: ∠A = ∠C.
Step 2: The second pair is also equal: ∠B = ∠D.
Step 3: Both pairs of opposite angles are equal.
Answer: Therefore, ABCD is a parallelogram.
Diagonals Bisect Each Other Test for a Parallelogram
AO = OC and BO = OD ⇒ ABCD is a parallelogram
SymbolMeaningUnit
AC, BDDiagonals of quadrilateral ABCDlength units
OIntersection point of the diagonalsNot applicable
AO, OCTwo parts of diagonal AClength units
BO, ODTwo parts of diagonal BDlength units
Worked example
Question: Diagonals AC and BD of quadrilateral ABCD intersect at O. If AO = OC = 4 cm and BO = OD = 3 cm, identify the quadrilateral.
Calculation:
Step 1: AO = OC, so O bisects diagonal AC.
Step 2: BO = OD, so O bisects diagonal BD.
Step 3: Therefore, the two diagonals bisect each other.
Answer: By the diagonal-bisection test, ABCD is a parallelogram.