Introduction: This is a key Geometry lesson. It's all about proving relationships ("riders"). The secret is to clearly state the theorem you are using in every step of your proof. Learn to identify the "same base" and the "parallel lines" in the diagrams.

The "S Pass" Foundation (නියත S එකකට)

• Prompt 1 (Recall): State the theorem about two parallelograms on the same base and between the same parallels.

• Prompt 2 (Recall): State the theorem about a parallelogram and a triangle on the same base and between the same parallels.

• Prompt 3 (Recall): State the theorem about two triangles on the same base and between the same parallels.

• Prompt 4 (Apply): In trapezium ABCD, AB || DC. What is the relationship between the areas of triangle ADC and triangle BDC? Why?

• Prompt 5 (Calculate): Parallelogram PQRS has an area of 80 cm². T is any point on the side SR. What is the area of triangle PQT?

  
  

Climbing to a "C" (C එකට පාර)

• Prompt 1 (Problem Solve): In the figure, ABCD is a rectangle and CDEF is a parallelogram. If AB = 10 cm and BC = 6 cm, find the area of parallelogram CDEF. State the theorem you used.

• Prompt 2 (Proof): The diagonals of trapezium PQRS intersect at X, with PQ || SR. Prove that Area(ΔPSR) = Area(ΔQSR).

• Prompt 3 (Deduction): Using the result from the previous prompt, prove that Area(ΔPXS) = Area(ΔQXR).

• Prompt 4 (Application): In triangle ABC, D is the midpoint of BC. Prove that Area(ΔABD) = Area(ΔACD).

• Prompt 5 (Proof): In parallelogram ABCD, P is any point on the diagonal BD. Prove that Area(ΔAPB) = Area(ΔCPB).

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Aiming for a "B" (B ඉලක්කය)

• Prompt 1 (Rider): ABCD is a parallelogram. A line through C parallel to DB meets AB produced at E. Prove that the area of triangle ADE is equal to the area of parallelogram ABCD.

• Prompt 2 (Rider): In triangle ABC, X is the midpoint of the median AD. Prove that Area(ΔABX) = 41​ Area(ΔABC).

• Prompt 3 (Rider): P is any point inside parallelogram ABCD. Prove that Area(ΔAPB) + Area(ΔDPC) = Area(ΔAPD) + Area(ΔBPC). (Hint: Draw a line through P parallel to AB).

• Prompt 4 (Application): E is the midpoint of the median AD of triangle ABC. The line BE produced meets AC at F. Prove that AC = 3AF. (Hint: Draw a line through D parallel to BF).

• Prompt 5 (Rider): In quadrilateral ABCD, a line through B parallel to AC meets DC produced at E. Prove that the area of triangle ADE is equal to the area of quadrilateral ABCD.

Securing the "A" Distinction (A සාමාර්ථය තහවුරු කරගන්න)

• Prompt 1 (Challenge Rider): The medians BE and CF of a triangle ABC intersect at G. Prove that the area of quadrilateral AFGE is equal to the area of triangle BGC.

• Prompt 2 (Challenge Rider): In parallelogram ABCD, O is any point on the diagonal AC. The line segment EOF is drawn parallel to AB, where E is on AD and F is on BC. The line segment GOH is drawn parallel to AD, where G is on AB and H is on DC. Prove that Area(parallelogram EG) = Area(parallelogram FH).

• Prompt 3 (Synthesis): ABCD and ABEF are two parallelograms on opposite sides of the common base AB. Prove that the area of parallelogram DCEF is the sum of the areas of parallelograms ABCD and ABEF.

• Prompt 4 (Construction Proof): Given a pentagon ABCDE, explain the steps to construct a triangle equal in area to the pentagon. Justify each step using the theorems from this lesson.

• Prompt 5 (Rider): The sides AB and AC of a triangle ABC are produced to P and Q respectively. The line PQ is parallel to BC. Prove that Area(ΔPC B) = Area(ΔQBC).