1. PSY-SOLUTION GEOMETRY (THEOREMS)

2. PSY-SOLUTION GEOMETRY (THEOREMS) Theorems in Mathematics are the established rules for further applications. Procedure of presenting a theorem is a logical, methodical and step-wise process. The steps to be followed religiously to have the total obtainment of markings. There are five steps, as follows: 1. Statements 2. Given 3. To prove 4. Construction 5. Proof.

3. PSY-SOLUTION 1.Statement: Mostly the statements are provided in the question itself, even though it is better to go through the statements properly and while answering it always good to start with the write-up of the statement. 2.Given: In this step we provide a figural presentation of the statement of the theorem. It consists the figure with alphabetical notation and its interpretations. The steps are as follows:-

4. PSY-SOLUTION 3.To Prove: In this step, we provide the only portion which we are going to prove strictly according to the figure. 4.Construction: In this step, we write about the constructions, we need to make, for proving the statement of the theorem. The constructions can be drawn in the same figure which we have already drawn in the ‘Given’ step.

5. PSY-SOLUTION 5.Proof: In this step, specifically we will prove the theorem. This step is conception-wise most important and we need to be brief and to the point. Remember: If in the process of proving a theorem we use another theorem for reference then we must mention the same in the bracket in the next to the respective step.

6. Let’s have an Example from book:-

7. Basic Proportionality Theorem Theorem 8.1 : If a line is drawn parallel to one side of a triangle, to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. Given : ∆ABC in which a line parallel to BC intersects AB at D and AC at E (Fig. 8.6) 6.1

8. Basic Proportionality Theorem (contd.)

9. PSY-SOLUTION List of theorems Theorem 6.1 : If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio Theorem 6.3 : If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar. Theorem 6.4 : If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of ) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar. Theorem 6.5 : If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar. Theorem 6.2 : If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

10. PSY-SOLUTION Theorem 6.7 : If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle Theorem 6.8 : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides and to each other. Theorem 6.9 : In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. Theorem 6.6 : The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

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