1. Each of the numbers 1 = 1, 3 = 1 + 2, 6 = 1+ 2+ 3, 10 = 1 + 2 + 3+4, ... represents the number of dots that can be arranged evenly in an equilateral triangle: This led the ancient Greeks to call a number triangular if it is the sum of consecutive integers, beginning with 1. Prove the following facts concerning triangular numbers: (a) A number is triangular if and only if it is of the form n(n + 1)/2 for some n > 1. (Pythagoras, circa 550 B.C.) (b) The integern is a triangular number if and only if 8n + 1 is a perfect square. (Plutarch, circa 100 A.D.) (c) The sum of any two consecutive triangular numbers is a perfect square. (Nicomachus, circa 100 A.D.) (d) If n is a triangular number, then so are 9n + 1, 25n + 3, and 49n + 6. (Euler, 1775) anouo thot in terms of the binomial coefficients,

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A: We will solve the given questions of the given triangle ABC  in the following step.

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A: (a)

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A: Note:- according to the company policy we can attempt only one question so please upload the next…

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A: A quadrilateral is a closed two dimensional shape having four vertices or four corners.

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Q: 28. The sum of all interior angles of a convex polygon of 'n' sides is equal to right angles.

A: The sum of all interior angles of a convex polygon of n sides is equal to ____ right angles

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