TrianglesTrianglesBy Eisa, Deva and Rajat

• Basic Proportionality Theorem• Similarity Criteria• Area Theorem• Pythagoras Theorem

What will you learn?

• 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

Proving the Thales’ Theorem

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

Proving the converse of Thales’ Theorem

Similarity Criteria

Similarity Criterias

SSS ASA AA

• 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

• 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

• 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

• 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

Proof of Area 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.

Pythagoras Theorem

Pythagoras Theorem

Proof 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.

Converse of Pythagoras Theorem

Proof of Converse of Pythagoras Theorem

• 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

• 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

• 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

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