For a polygon to be convex means that given any two points on or inside the polygon, the line join- ing the points lies entirely inside the polygon. Use mathematical induction to prove that for every integer n > 3, the angles of any n-sided convex polygon add up to 180(n- 2) degrees

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A: To answer the given questions we shall use the following definition  An integer n is said to be even…

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A: A polygon is a geometrical figure that consists of more than one side.  When it contains n sides, it…

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Q: In Exercises 35 and 36, use Theorem 2.5.1 to solve the prob- lem. According to this theorem, the…

A: Definition used -                     Number of diagonals -                          D= n(n-3)2…

Q: If a polygon is convex, then all its points must lie inside the interior of each of its vertex…

A: Properties of the polygon To find which are true and to give examples.

Q: Suppose that the side lengths of one of a pair of 5-con trianglesare given by the numbers (a, b, c),…

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Q: Given the universe of discourse as the set of natural numbers, N, use induction to prove P(n): 2"…

A: Here prove that Pn: 2n>n2 is true ∀n≥m

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Q: The general combinatorial formula for the nth triangular number (1, 3, 6, 10, 15, 21, .) is On+1Cn…

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Q: .Prove that the sum of the cabes of first A s cqual to the positive integers is quare of the cgad…

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Q: 2. Find the area of the regular pentagon with apothem 2.8 and side 4. Not drawn to scale. 4 in. 2.8…

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Q: Given: NPQRST is a regular hexagon. Prove: NPRS is a rectangle. (Begin by drawing a diagram.)

A: To prove: NPRS is a rectangle. Concept used: All the sides and all angles of Regular hexagon are…

Q: Theorem 3: Ifconverges to l, ther any subsequence ofalso comverges to l.

A: We have given that sn converges to l. Let snk be any subsequence of sn. We need to prove that snk…

Q: Prove that, if a is an irrational number, then Va is also an irrational number.

A: In the given problem we have to prove that if x is an irrational number, then x3 is also an…

Q: Show that each conjecture is false by finding a counterexample. 2. For any integer n, n° > 0. 3.…

A: # 2) given statement is n 3 > 0 , for any integer n

Q: 2. The Polygon Exterior Angle Sum Theorem states the sum of the measures of the exterior angles of a…

A: We have to calculate the sum of exterior angles of polygon.

Q: A diagonal in a polygon is a line segment that is drawn from a vertex to a nonadjacent vertex. If n…

A: Polygon has n sides and therefore n vertices. So total possible lines we can make = nC2 Out of which…

Q: Prove that the Ving of ComPelX numbers C¢,+.)is fielol

A: Solution is given as below:

Q: or an integer n>=3, an n-gon is an n-sided polygon. (example, a 3-gon is a triangle, 4-gon is a…

A: To prove: the sum of the interior anles of every polygon with n sides is n-2180° Now we shall take a…

Q: 3. Prove the following two theorems about pairs of "twin primes," p and (Recall that "twin primes"…

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Q: on 'N" en Prove Delween Aborms is lenre equala bub

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Q: A convex quadrilateral has area 30 and side lengths 5, 6, 9 and 7, in that order. Denote by θ the…

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Q: Show that for any nez, n is erther odd or even

A: Suppose P(n) is a given statement, then P(n) is true for all positive integer if: P(1) is true. If…

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Q: Prove that for all primes p2 5, p + 1 cannot be a perfect square. Recall that is a perfect square if…

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Q: Prove by induction that the expression for the number of diagonals in a polygon of n sides is…

A: Solution: Let Dn=nn-32 We will prove by induction that for all n∈ℕ, the total number of diagonals in…

Q: Number theory.

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Q: A pentagonal pyramidal number P can be determined using the dot representations of pentagonal…

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Q: 12. Prove that for all integers a and m, if a and m are the lengths of the sides of a right triangle…

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Q: Let n 2 3 be a natural number. Find how many pairwise non-congruent triangles are there among the ("…

A: Given that n≥3. Now consider regular n-gon.  It has partition of n edges into 3 pieces with x edges…

Q: . Use direct proof to prove that, if x is a positive odd integer, then z – 1 is wholly dividable у…

A: x^2-1 is divisible by 4.

Q: 4. Prove that the cube of any integer can be written as the difference of Notice that squ n³ = (1³ +…

A: To prove that cube of any integer can be written as the difference of two sqaures.we have given hint…

Q: In the figure below, the shape is completed by combining a regular hexagon and a square with its…

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Q: How many symmetries does a regular n-gon have? (This means an n-sided polygon, all of whose sides…

A: We find number of symmetries of triangle, square, pentagon.

Q: In ANOP, NP is extended through point P to point Q, mZNOP= (3x + 8)°, MZPNO = (2x + 9)°, and mZOPQ =…

A: To find the value of x

Q: 19. Using only the digits 1 through 9 one time each, is it possible to construct a 3 by 3 magic…

A: A magic square of order n is  an arrangement of an array of order n×n in which the sum of numbers in…

Q: Let a and b be positive integers and let d = gcd(a, b) and m =lcm(a, b). If t divides both a and b,…

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Q: Prove using weak induction that: For any convex polygon with n sides (n ≥3), any triangulation…

A: now if we look at this procedure This is strong induction. Let Φ(n) be the statement that all…

Q: the pth term of AP is a and the qth term is p,prove that it's nth term is (p+q-n)

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Q: Suppose that p is a prime number greater than 30. Prove that when p is divided by 30, the remainder…

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Q: Find a polynomich of Jegree 4 with Zeros x = 1₁ x=2, and x=i.

A: Given polynomial has zeros x=1, x=2 and x=i. Therefore (x-1) ,(x-2) and (x-i) is factor of…

Q: All sides in a regular polygon must be congruent. To True O False

A: A polygon which has all sides mutually congruent and all angles mutually congruent is called a…

Q: ny regular polygon, the center can be determined by the intersection of any two angle-bisectors of…

A: Given: For any regular polygon, the center can be determined by the intersection of any two…

Q: Prove that, if z is an irrational number, then Va is also an irrational number.

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Q: Suppose that and K are points on the number line. IF JK = 13 and lies at 5, where could K be…

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Q: Prove that If p is prime, p divides k^2 + m^2 and p divides m^2 + x^2, then p divides k^2 + x^2

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Q: Prove by Mathematical Induction, for all integers n 2 3, that the sum of the interior angles of ar…

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