answer:In a kite, the properties dictate that adjacent pairs are congruent (VW = WY and VX = YX). Additionally, the angles between unequal sides are equal, specifically angles V and Y for this kite. Since it is given that angles V and Y are right angles, they both measure 90 degrees. In a kite, the angles adjacent to each right angle are supplementary to the right angle because the sum of angles around a point is 360 degrees and the pairs of angles between the same two sides must add up to 180 degrees, given the kite's symmetry. Since angle W is given as (18x - 2) degrees and angle X is given as (12x + 2) degrees, we can find the value of x by setting an equation that the sum of angle W and angle X equals 180 degrees, because they are both supplementary to angle V or angle Y. Setting up the equation: (18x - 2) + (12x + 2) = 180 Now combine like terms: 18x + 12x = 180 30x = 180 Divide both sides by 30 to solve for x: x = 180 / 30 x = 6 Now that we have the value of x, we can use it to find the measure of angle X: Angle X = 12x + 2 Angle X = 12(6) + 2 Angle X = 72 + 2 Angle X = 74 degrees Therefore, the measure of angle VXY, which is the same as angle X, is 74 degrees. The correct answer is: 74o

answer:Given that we have a quadrilateral with 90-degree angles, we can infer that it is a rectangle because a rectangle is the only quadrilateral that has four right angles. In a rectangle, opposite sides are equal in length. The question tells us: - The lengths of the left and right sides are 10 and 2y + 4. - The lengths of the top and bottom sides are 18 and 10 + 2x. Since opposite sides of a rectangle are equal, we can set the lengths of the top and bottom sides equal to each other: 18 = 10 + 2x Now we solve for x: 18 - 10 = 2x 8 = 2x Divide both sides by 2 to get the value of x: x = 8 / 2 x = 4 Therefore, the value of x is 4. The correct answer is: 4

answer:Given that the quadrilateral has 90-degree angles, it is a rectangle, where opposite sides are of equal length. The lengths of the left and right sides are given as 10 and 2y + 4, so because these are opposite sides of a rectangle, they should be equal in length. Therefore: 10 = 2y + 4 To find the value of y, we subtract 4 from both sides of the equation: 10 - 4 = 2y 6 = 2y Now, divide both sides by 2 to solve for y: 6 / 2 = y 3 = y So the value of y is 3. The correct answer is: 3

answer:The provided "proof" falsely claims to demonstrate that any real number a raised to any natural number power n equals 1, when in fact this is only true for a = 1. Let's dissect the given "proof" and identify where the error lies. Base Case: The proof correctly states that a^0 = 1 for all nonzero real numbers a. This is a universally accepted mathematical truth based on the definition of exponents. Inductive Hypothesis: The proof then assumes as the inductive hypothesis that a^n = 1 for all integers 0 ≤ n ≤ k. This is a standard step in an inductive proof, where we assume the statement is true for a certain range of integers. Inductive Step: Here, the "proof" tries to show that if the statement is true for all integers up to k, it is also true for k + 1. The error occurs in how the proof is attempting to demonstrate this. The "proof" manipulates the expression a^(k+1) by writing it incorrectly as a^(k+1) · a^(k−1) and then as (a^k · a^k) / (a^(k−1) · a^(k−1)). The expression a^(k+1) = a^(k+1) · a^(k−1) does not make mathematical sense and is unrelated to the sought identity a^(k+1) = a^k * a. Therefore, this transformation is invalid. Additionally, even if we were to overlook this mistake, the false "proof" wrongly assumes that a^(k−1) also equals 1 by the inductive hypothesis. However, the inductive hypothesis only assumed that the statement was true up to k, not k - 1. So, one cannot simply invoke the hypothesis to assert that a^(k−1) equals 1. Returning to the phrase "Therefore, the proof fails for k = 0, as this would need a^(−1) = 0", it seems to be another mistake, albeit a typographical one. It should actually say "this would need a^(−1), which was neither proven nor true" — in other words, there's no established base case for a^1 or any mechanism for the proof to handle inverse (negative) powers such as a^(-1). The mention of a^(−1) = 0 appears to be an error in the explanation; the critical thing is that a^(-1) is required to make the faulty inductive step logic work, but it isn't part of the induction hypothesis. In summary, the main flaw in the "proof" lies in incorrect manipulation of exponents during the inductive step. Furthermore, the explanation contains a typographical error concerning a^(-1), which should not equate to 0 since a^(-1) is defined as 1/a, and for it to equal 0 would violate the fundamental rule that division by zero is undefined.