This lesson introduces a key geometric shape: the parallelogram. The focus is on understanding its fundamental properties and using them to solve problems. You start with the knowledge that a shape is a parallelogram and use its special rules to find unknown lengths and angles.

1. Core Concepts & Properties

• The Definition: A parallelogram is a quadrilateral where both pairs of opposite sides are parallel. This single rule is the foundation for all other properties.

  1. In parallelogram ABCD: AB || DC and AD || BC.

• The Four Key Properties You Must Know:

  1. Opposite sides are equal. (AB = DC and AD = BC)

  2. Opposite angles are equal. (∠A = ∠C and ∠B = ∠D)

  3. Diagonals bisect each other. (They cut each other exactly in half at the point of intersection). If diagonals AC and BD intersect at O, then AO = OC and BO = OD.

  4. Each diagonal bisects the area. (A diagonal splits the parallelogram into two congruent triangles of equal area).

2. A Step-by-Step Strategy for Solving Problems

When you see a parallelogram in an exam, use this mental checklist to find what you need.

For Calculation Problems (Finding lengths or angles):

1. Think Parallel Lines First: The parallel sides (AB || DC and AD || BC) are your most powerful tool. Look for:

   • Alternate Interior Angles (Z-shape): These are equal.

   • Consecutive Interior Angles (C-shape): These are supplementary (add up to 180°). For example, ∠A + ∠B = 180°.

2. Use the Properties as Shortcuts:

   • Need a side length? Look at the side opposite it. They are equal.

   • Need an angle? Look at the angle opposite it. They are equal. Or look at the angle next to it; they add up to 180°.

   • Dealing with diagonals? If you know the length of half a diagonal (e.g., AO), you know the other half (OC) is the same.

For Proof (Rider) Problems:

The most common strategy is to prove that two triangles inside the parallelogram are congruent.

1. Identify the Goal: What exactly do you need to prove? (e.g., Prove that AQ = PC).

2. Find the Key Triangles: Locate two triangles that contain the sides or angles you need to prove equal. (e.g., For AQ and PC, the triangles ΔADQ and ΔCBP are the obvious choice).

3. Prove the Triangles are Congruent: Use the properties of the parallelogram to get your three conditions (e.g., SAS, AAS).

   • You can get Sides from "Opposite sides are equal".

   • You can get Angles from "Opposite angles are equal" or "Alternate angles are equal" (using the parallel sides).

4. State Your Conclusion: Once the triangles are congruent, you can state that the parts you needed are equal because they are "Corresponding sides/angles of congruent triangles".

Example Proof:

• To Prove: ΔADQ ≅ ΔCBP in a parallelogram ABCD where P and Q are points on the diagonal BD.

• Proof:

  • AD = CB (Opposite sides of a parallelogram are equal) -> Side

  • ∠ADQ = ∠CBP (Alternate angles, since AD || BC) -> Angle

  • Assume we are given DQ = BP -> Side

  • Therefore, ΔADQ ≅ ΔCBP (SAS).

4. Important Points to Remember

• Give Reasons! For every geometric statement in a proof, you must write the correct reason in brackets.

  • (Opposite sides of a parallelogram)

  • (Alternate angles, AD || BC)

  • (Diagonals of a parallelogram bisect each other)

• Don't Assume Properties of Special Parallelograms: A general parallelogram does not have equal diagonals or 90° angles. Only use the four properties listed above unless you are told it is a rectangle or a square.