This is a key geometry lesson for your exam. It's not about calculating a specific area with numbers, but about proving that the areas of different shapes are equal or related in a specific way (e.g., one is half of the other). The entire lesson is built on three core theorems.

1. The Core Idea: Same Base, Same Height

All the theorems in this lesson work on a simple principle: If two shapes share the exact same base and are drawn between the same pair of parallel lines, they must have the same perpendicular height.

Area of Parallelogram = base × height Area of Triangle = ½ × base × height

Since the base and height are the same, their areas will be directly related.

2. The Three Key Theorems

Theorem 1: Parallelograms on the Same Base

"Parallelograms on the same base and between the same parallels are equal in area."

• In simple terms: If you have two "slanted" parallelograms sharing the same bottom line and their top lines are on the same parallel line, their areas are identical.

• Why? They share the same base (b) and the same perpendicular height (h). So, Area = b × h is the same for both.

Theorem 2: Parallelogram and Triangle on the Same Base

"If a parallelogram and a triangle are on the same base and between the same parallels, the area of the triangle is half the area of the parallelogram."

• In simple terms: A triangle on the same base and between the same parallel lines as a parallelogram will have exactly half the area.

• Why? The parallelogram's area is b × h. The triangle's area is ½ × b × h.

Theorem 3: Triangles on the Same Base

"Triangles on the same base and between the same parallels are equal in area."

• In simple terms: Any two triangles that share the same base and have their third vertex on the same parallel line will have equal areas.

• Why? Both have the same base (b) and height (h), so Area = ½ × b × h is the same for both.

How to Solve Exam Problems (Riders)

Most exam questions will ask you to prove that two areas are equal. Follow this exact structure in your answer:

1. Identify the two larger shapes you are comparing.

2. State the Common Base: "Parallelogram ABCD and Triangle ABX share the common base AB."

3. State the Parallel Lines: "...and lie between the same parallels AB and DC."

4. State the Conclusion and the Theorem: "Therefore, Area of ΔABX = ½ Area of ||gm ABCD (Theorem: A parallelogram and a triangle on the same base...)"

Exam Tips & Common Rider Tricks

• The "Subtracting the Common Area" Trick: This is one of the most common exam questions. You will be given a trapezium ABCD and asked to prove that the diagonal triangles are equal in area (prove Area of ΔAOD = Area of ΔBOC).

  • How to solve it:

    1. First, prove two larger triangles are equal: Area of ΔADC = Area of ΔBDC (They have a common base DC and are between parallels AB and DC).

    2. Then, state that you are subtracting the common area from both: Area of ΔADC - Area of ΔDOC = Area of ΔBDC - Area of ΔDOC.

    3. This leaves you with the result: Area of ΔAOD = Area of ΔBOC.

• Tip 1: Find the Parallels First. When you get a problem, the very first thing you should do is identify the pair of parallel lines. This tells you which shapes you can apply the theorems to.

• Tip 2: Look for the Common Base. After finding the parallel lines, look for shapes that share a side on one of those lines. This is your "common base".

• Mistake to Avoid: Don't confuse "same base" with "equal base". The shapes must be built on the exact same line segment. Two separate but equal line segments do not count.