This document discusses various theorems and properties related to triangles. It explains the Basic Proportionality Theorem, also known as Thales' Theorem, which states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. It also covers similarity criteria like AAA, SSA, and SSS. The Area Theorem demonstrates that the ratio of areas of similar triangles equals the square of the ratio of corresponding sides. Additionally, it proves Pythagoras' Theorem, which relates the sides of a right triangle, and its converse. In summary, the document outlines key triangle theorems regarding proportional division, similarity, areas, and the Pythagorean relationship between sides.

1. TrianglesTriangles
By Eisa, Deva and Rajat

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. ProvingtheconverseofThales’
Theorem

7. Similarity Criteria
Similarity Criterias
SSS ASA AA

8. • 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.
AAA Similarity

9. • 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.
SSS Similarity

10. • 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.
SAS Similarity

11. • 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
Area Theorem

12. ProofofAreaTheorem

13. • 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.
Pythagoras Theorem

14. Pythagoras Theorem

15. ProofofPythagorasTheorem

16. 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.
Converse of Pythagoras Theorem

17. ProofofConverseofPythagorasTheorem

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

21. Thank you