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Geometry and Trigonometry Difficulty: Hard

A line intersects two parallel lines, forming four acute angles and four obtuse angles. The measure of one of these eight angles is $left parenthesis 7 x minus 250 right parenthesis degree$ . The sum of the measures of four of the eight angles is $k °$ . Which of the following could NOT be equivalent to $k$ , for all values of $x$ ?

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Explanation

Choice A is correct. It’s given that a line intersects two parallel lines, forming four acute angles and four obtuse angles. Since there are two parallel lines intersected by a transversal, all four acute angles have the same measure and all four obtuse angles have the same measure. Additionally, each acute angle is supplementary to each obtuse angle. It’s given that the measure of one of these eight angles is $left parenthesis 7 x minus 250 right parenthesis degree$ . It follows that a supplementary angle has measure $(180 - (7 x - 250)) °$ , or $(- 7 x + 430) °$ . It’s also given that the sum of the measures of four of the eight angles is $k °$ . It follows that the possible values of $k$ are $4 left parenthesis 7 x minus 250 right parenthesis$ ; $(7 x - 250) + 3 (- 7 x + 430)$ ; $2 (7 x - 250) + 2 (- 7 x + 430)$ ; $3 (7 x - 250) + (- 7 x + 430)$ ; and $4 (- 7 x + 430)$ . These values are equivalent to $28 x minus 1,000$ ; $- 14 x + 1,040$ ; $360$ ; $14 x minus 320$ ; and $- 28 x + 1,720$ , respectively. It follows that of the given choices, only $- 14 x + 1,540$ could NOT be equivalent to $k$ , for all values of $x$ .

Choice B is incorrect. This is the sum of three angles with measure $left parenthesis 7 x minus 250 right parenthesis degree$ and one angle with measure $(- 7 x + 430) °$ .

Choice C is incorrect. This is the sum of four angles with measure $(- 7 x + 430) °$ .

Choice D is incorrect. This is the sum of two angles with measure $left parenthesis 7 x minus 250 right parenthesis degree$ and two angles with measure $(- 7 x + 430) °$ .