Icse Parallelogram Proofs
Geometry — Icse Parallelogram Proofs
Icse Parallelogram Proofs
Parallelogram Proofs (ICSE)
Key Properties of a Parallelogram
A parallelogram is a quadrilateral with two pairs of parallel opposite sides.
| Property | Statement |
|---|---|
| P1 | Opposite sides are equal: AB = CD, AD = BC |
| P2 | Opposite angles are equal: ∠A = ∠C, ∠B = ∠D |
| P3 | Adjacent angles are supplementary: ∠A + ∠B = 180° |
| P4 | Diagonals bisect each other: AO = OC, BO = OD (O = intersection) |
| P5 | Each diagonal divides the parallelogram into two congruent triangles |
Proof 1: Opposite Sides are Equal (P1)
Given: ABCD is a parallelogram (AB ∥ CD, AD ∥ BC) To prove: AB = CD and AD = BC
Construction: Draw diagonal AC.
Proof: In △ABC and △CDA:
- ∠BAC = ∠DCA (alternate interior angles, AB ∥ DC, AC is transversal)
- ∠BCA = ∠DAC (alternate interior angles, BC ∥ AD, AC is transversal)
- AC = AC (common side)
∴ △ABC ≅ △CDA (by ASA) ∴ AB = CD and BC = AD (CPCT) QED
Proof 2: Diagonals Bisect Each Other (P4)
Given: ABCD is a parallelogram, diagonals AC and BD intersect at O To prove: AO = OC and BO = OD
In △AOB and △COD:
- ∠OAB = ∠OCD (alternate angles, AB ∥ CD)
- ∠OBA = ∠ODC (alternate angles, AB ∥ CD)
- AB = CD (proved above)
∴ △AOB ≅ △COD (by ASA) ∴ AO = OC and BO = OD (CPCT) QED
Proof 3: Converse — If Diagonals Bisect Each Other, It's a Parallelogram
Given: Quadrilateral ABCD with diagonals bisecting each other at O (AO = OC, BO = OD) To prove: ABCD is a parallelogram
In △AOB and △COD:
- AO = OC (given)
- BO = OD (given)
- ∠AOB = ∠COD (vertically opposite angles)
∴ △AOB ≅ △COD (SAS) ∴ AB = CD and ∠OAB = ∠OCD → AB ∥ CD
Similarly, △AOD ≅ △COB → AD = BC and AD ∥ BC ∴ ABCD is a parallelogram. QED
Special Parallelograms
| Shape | Extra Property |
|---|---|
| Rectangle | All angles = 90°; diagonals equal |
| Rhombus | All sides equal; diagonals perpendicular bisectors |
| Square | All sides equal + all angles 90°; diagonals equal + perpendicular |
ICSE Proof Technique Tips
- State given information clearly
- Identify which triangles to prove congruent
- Name the congruence criterion: SSS, SAS, ASA, AAS, RHS
- Use CPCT (corresponding parts of congruent triangles) to extract what you need
- Write QED (quod erat demonstrandum) at the end
Standard ICSE Proof Question Types
- Prove that the diagonals of a rhombus bisect each other at right angles
- In parallelogram ABCD, E and F are midpoints of AB and CD — prove AFCE is a parallelogram
- Prove that the line segment joining midpoints of two sides of a triangle is parallel to the third side (Midpoint Theorem)
Quick Check
- In parallelogram PQRS, ∠P = 70°. Find all other angles.
- If diagonals of a quadrilateral bisect each other, what can you conclude? Why?
- In rhombus ABCD, the diagonals meet at O. Prove that ∠AOB = 90°.
- State the condition that makes a parallelogram a rectangle.
- Stretch: ABCD is a parallelogram and E is the midpoint of AB. DE extended meets BC extended at F. Prove that DF = 2 × DE.
Key Takeaways (TL;DR)
- Key Properties of a Parallelogram
- Proof 1: Opposite Sides are Equal (P1)
- Proof 2: Diagonals Bisect Each Other (P4)
- Proof 3: Converse — If Diagonals Bisect Each Other, It's a Parallelogram
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