Introduction: This is another major geometry topic. The key to success is correctly identifying the two similar triangles and then, most importantly, matching their corresponding sides to set up the correct ratios. Always write the triangles out side-by-side to match the vertices.

The "S Pass" Foundation (නියත S එකකට)

• Prompt 1 (State): State the theorem about a line drawn parallel to one side of a triangle.

• Prompt 2 (Apply): In ΔABC, DE || BC. If AD=3, DB=6, and AE=4, find EC.

• Prompt 3 (State): What are the two conditions for two triangles to be similar (equiangular)?

• Prompt 4 (Apply): ΔPQR ~ ΔXYZ. If PQ=5, QR=6, and XY=7.5, find YZ.

• Prompt 5 (Identify): In ΔABC, B^=90∘ and BD is the altitude to AC. Name three similar triangles in the diagram.

Climbing to a "C" (C එකට පාර)

• Prompt 1 (Problem Solve): In ΔPQR, S is on PQ and T is on PR such that ST || QR. If PS=4, SQ=2 and TR=3, find PT.

• Prompt 2 (Proof): In trapezium ABCD with AB || DC, the diagonals intersect at O. Prove that ΔAOB is similar to ΔCOD.

• Prompt 3 (Problem Solve): A boy of height 1.5 m stands 12 m away from a flagpole. His shadow is 3 m long. How tall is the flagpole?

• Prompt 4 (Proof): The angle bisector of angle A in ΔABC meets BC at D. A line through D parallel to AB meets AC at E. Prove that ΔCDE is similar to ΔCBA.

• Prompt 5 (Application): State and prove the converse of the proportionality theorem.

Aiming for a "B" (B ඉලක්කය)

• Prompt 1 (Rider): In ΔABC, the point D lies on BC. The line drawn from D, parallel to AB meets AC at E. The line drawn from D parallel to AC meets AB at F. Prove that ECAE​=FABF​.

• Prompt 2 (Rider): P is any point inside a parallelogram ABCD. The line passing through P parallel to AB meets AD at Q and BC at R. The line passing through P parallel to AD meets AB at S and DC at T. Prove that ΔQPS is similar to ΔTRC.

• Prompt 3 (Area Ratio): If two triangles are similar, prove that the ratio of their areas is equal to the square of the ratio of their corresponding sides.

• Prompt 4 (Application): Using the result from the previous prompt, if ΔABC ~ ΔPQR, Area(ABC)=64cm² and Area(PQR)=100cm², and AB=4cm, find PQ.

• Prompt 5 (Rider): In a right-angled triangle ABC, B^=90∘. BD is the perpendicular from B to AC. Prove that AB2=AD⋅AC.

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

• Prompt 1 (Challenge Rider): In cyclic quadrilateral ABCD, the diagonals AC and BD intersect at P. Prove that (i) ΔAPB is similar to ΔDPC, and (ii) AP⋅PC=BP⋅PD.

• Prompt 2 (Challenge Rider): The angle bisector of angle A in triangle ABC meets BC at D. Prove the Angle Bisector Theorem: ACAB​=DCBD​. (Hint: Draw a line through C parallel to AD to meet BA produced).

• Prompt 3 (Synthesis): A circle passes through the vertex C of a rectangle ABCD and touches the sides AB and AD at points M and N respectively. If CM produced intersects AD produced at P, prove that BM⋅DN=AB⋅AD.

• Prompt 4 (Rider): In a parallelogram PQRS, the side PQ is produced to any point T. The line TS intersects PR at A and QR at B. Prove that TA2=AB⋅AT.

• Prompt 5 (Proof): Provide a formal proof for the theorem: "If the three sides of a triangle are proportional to the three sides of another triangle, then the two triangles are equiangular."