Question

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Triangles chapter

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Class 10

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➡Please give Important theorems and Important questions

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Answer 1

$\large \huge{ \mathfrak{ \star \star HELLO \: FRIENDS!! \star \star}}$

$\huge \underline{Here \: is \: your \: answer}$

$\huge \boxed{Important \: theorems.}$

1. If a line is drawn parallel to one side of a triangle to intersect the other two side in distinct points then the other two sides are divided in the same ratio. [ THALES' THEOREM OR BASIC - PROPORTIONALITY THEOREM ].

2. IF A LINE DIVIDES ANY TWO SIDES OF A TRIANGLE IN THE SAME RATIO THEN THE LINE MUST BE PARALLEL TO THE THIRD SIDE. [ Converse of Thales' theorem ].

3. The line segment joining the midpoint of any two side of a triangle is parallel to the third side. [ mid-point theorem ].

4. The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.
[ Angle - Bisector theorem ].

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$\boxed{Similarity \: Criteria.}$

5. ( AAA - Similarity ) If in two triangles, the corresponding angles are equal, then their corresponding sides are proportional and hence the triangles are similar.

6. ( SSS - Similarity ) If the corresponding sides of two Triangles are proportional then their corresponding angles are equal, and hence the two Triangles are similar.

7. ( SAS - Similarity ) 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.

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8. The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

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9. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. [ PYTHAGORAS' THEOREM ].

10. IN A TRIANGLE, IF THE SQUARES OF ONE SIDE IS EQUAL TO THE SUM OF THE SQUARES OF THE OTHER TWO SIDES THEN THE ANGLE OPPOSITE TO THE FIRST SIDE IS A RIGHT ANGLE. [ Converse of Pythagoras' theorem ].

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$\huge \boxed{Important \: questions :-)}$

1. ABCD is a trapezium in which AB || CD and it's diagonal intersect each other at the point O.
prove that :- (AO/OC) = (BO/OD).

2. Prove that the line segment joining the midpoints of adjacent sides of the quadrilateral form of parallelogram.

3. If the bisector of an angle of a triangle bisects the opposite side, prove that the triangle is isosceles.

4. If the diagonal BD of a quadrilateral ABCD bisects both <B and <D ,
prove that :- (AB/BC) = (AD/CD).

5. Prove that if a perpendicular is drawn from the vertex of the right angle of a triangle to the hypotenuse then the triangle on both sides of the perpendicular are similar to the whole triangle and also to each other.

6. Prove that if two angles of one triangle are respectively equal to two angles of another triangle then the two triangles are similar.

7. If two triangles are equiangular, prove that the ratio of their corresponding sides is the same as the ratio of the corresponding altitudes.

8. If two triangles are equiangular, prove that the ratio of their corresponding sides is the same as the ratio of the corresponding medians.

9. If two triangles are equiangular, show that the ratio of the corresponding sides is the same as the ratio of the cross ponding angle bisector segments.

10. Prove that the ratio of the perimeter of two similar triangles is same as the ratio of their corresponding sides.

11. Diagonals AC and BD of a trapezium ABCD with AB || CD intersect each other at point O. Using a similarity criterion for two triangles, show that (OA/OC) = (OB/OD).

12. Prove that the line segments joining the midpoints of the sides of a triangle form four triangles, each of which is similar to the original triangle.

13. If the areas of two similar triangles are equal then prove that the Triangles are congruent.

14. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding altitudes.

15. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding medians.

16. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding angle-bisector segments.

17. Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of an equilateral triangle described on one of its diagonals.

18. Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonal.

19. Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Prove that ∆ABC similar to ∆PQR.

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$\huge \boxed{THANKS}$

$\huge \bf \underline{Hope \: it \: is \: helpful \: for \: you}$