This Power point Presentation will give you knowledge about triangles.

2. Basic Proportionality Theorem
Similarity Criteria
Area Theorem
Pythagoras Theorem
What will you
learn?

3.  Basic Proportionality Theorem states that if a line is
drawn parallel to one side of a triangle to intersect the
other 2 points , the other 2 sides are divided in the same
ratio.
 It was discovered by Thales , so also known as Thales
theorem.
Basic Proportionality
Theorem

4. ProvingtheThales’
Theorem

5. Converse of the Thales’
Theorem
If a line divides any two sides of a
triangle in the same ratio, then the line
is parallel to the third side

6. Provingtheconverseof
Thales’Theorem

7. Similarity
Criteria
Similarity Criterias
SSS ASA AA

8. AAA
Similarity
 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.
 In Δ ABC and Δ DEF if ∠ A=∠ D, ∠ B= ∠E and ∠ C =∠ F then
Δ ABC ~ Δ DEF.

9. SSS Similarity
 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.
 In Δ ABC and Δ DEF if AB/DE =BC/EF =CA/FD then Δ
ABC ~ Δ DEF.

10. 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.
 In Δ ABC and Δ DEF if AB/DE =BC/EF and ∠ B= ∠E
then Δ ABC ~ Δ DEF.

11. Area Theorem
 The ratio of the areas of two similar triangles is equal
to the square of the ratio of their corresponding sides
 It proves that in the figure given below

12. ProofofArea
Theorem

13. Pythagoras
Theorem
 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 and to each other
 In a right triangle, the square of the hypotenuse is equal
to the sum of the squares of the other two sides.
 In a right triangle if a and b are the lengths of the legs
and c is the length of hypotenuse, then a² + b² = c².
 It states Hypotenuse² = Base² + Altitude².
 A scientist named Pythagoras discovered the theorem,
hence came to be known as Pythagoras Theorem.

15. ProofofPythagoras
Theorem

16. Converse of Pythagoras
Theorem
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.

17. ProofofConverseofPythagoras
Theorem

18.  Two figures having the same shape but not necessarily the same size
are called similar figures.
 All the congruent figures are similar but the converse is not true.
 Two polygons of the same number of sides are similar, if (i) their
corresponding angles are equal and (ii) their corresponding sides
are in the same ratio (i.e., proportion).
 If a line is drawn parallel to one side of a triangle to intersect the
other two sides in distinct points, then the other two sides are
divided in the same ratio.
 If a line divides any two sides of a triangle in the same ratio,
then the line is parallel to the third side.
Summary

19.  If in two triangles, corresponding angles are equal, then their
corresponding sides are in the same ratio and hence the two
triangles are similar (AAA similarity criterion).
 If in two triangles, two angles of one triangle are respectively
equal to the two angles of the other triangle, then the two
triangles are similar (AA similarity criterion).
 If in two triangles, corresponding sides are in the same ratio,
then their corresponding angles are equal and hence the
triangles are similar (SSS similarity criterion).
 If one angle of a triangle is equal to one angle of another
triangle and the sides including these angles are in the same
ratio (proportional), then the triangles are similar (SAS
similarity criterion).
 The ratio of the areas of two similar triangles is equal to the
square of the ratio of their corresponding sides.
Summary

20.  If a perpendicular is drawn from the vertex of the
right angle of a right triangle to the hypotenuse, then
the triangles on both sides of the perpendicular are
similar to the whole triangle and also to each other.
 In a right triangle, the square of the hypotenuse is
equal to the sum of the squares of the other two
sides (Pythagoras Theorem).
 If in a triangle, 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.
Summary