This document contains 16 sections that summarize various theorems about triangle similarity. Theorem 6.1 states if a line is drawn parallel to one side of a triangle to intersect the other two sides, the two sides are divided in the same ratio. Theorem 6.3 states if two triangles have corresponding angles that are equal, then the corresponding sides are in the same proportion and the triangles are similar. Theorem 6.7 states if a perpendicular is drawn from the right angle vertex to the hypotenuse of a right triangle, the triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

1. VISHAL JHA
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2. Table of content
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3. THEOREM 6.1
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5. THEROREM 6.2
THEOREM 6.2- If a line is drawn parallel to one side of a triangle to intersect the
two other side in distinct points , the other two side are divided in the same
ratio
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6. THEOREM 6.3
Theorem 6.3-If two triangles, corresponding angles are equal , then their
corresponding sides are in the same proportion and hence the two triangles are
similar
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7. Theorem 6.4
• 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.
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8. Theorem 6.5
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
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9. Theorem 6.6
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11. 𝑎𝑟(𝐴𝐵𝐶)
𝑎𝑟(𝑃𝑄𝑅)
= 𝐴𝐵 𝑃𝑄 2 = 𝐵𝐶 𝑄𝑅 2 = 𝐶𝐴 𝑅𝑃 2
Theorem 6.7
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 and to each other.
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12. Theorem 6.8
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14. THEOREM 6.9
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15. Now in triangle PQR , we have
PR2 = PQ2 +QR2 (PythagorasTheorem as∠𝑄 = 90° )
PR2 =AB2 + BC2 (By construction) (1)
But AC2 = AB2 +BC2 (Given) (2)
So, AC=PR [From (1)and (2)] (3)
Now In triangles ABC andPQR,
AB = PQ (By construction)
BC = QR (By construction)
AC = PR (Proved above in 3)
ABC ≅ PQR (SSS congruence )
∠𝐵 = ∠𝑄 (By CPCT)
∠𝐵 = ∠𝑄=90°
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16. THANKYOU
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