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Geometry and Trigonometry Difficulty: Medium

In the figure shown, lines $r$ and $s$ are parallel, and line $m$ intersects both lines. If $y less than 65$ , which of the following must be true?

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Explanation

Choice B is correct. In the figure shown, the angle measuring $y °$ is congruent to its vertical angle formed by lines $s$ and $m$ , so the measure of the vertical angle is also $y °$ . The vertical angle forms a same-side interior angle pair with the angle measuring $x °$ . It's given that lines $r$ and $s$ are parallel. Therefore, same-side interior angles in the figure are supplementary, which means the sum of the measure of the vertical angle and the measure of the angle measuring $x °$ is $180 degree$ , or $x + y = 180$ . Subtracting $x$ from both sides of this equation yields $y = 180 - x$ . Substituting $180 minus x$ for $y$ in the inequality $y less than 65$ yields $180 minus x less than 65$ . Adding $x$ to both sides of this inequality yields $180 less than 65 plus x$ . Subtracting $65$ from both sides of this inequality yields $115 less than x$ , or $x greater than 115$ . Thus, if $y less than 65$ , it must be true that $x greater than 115$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. $x + y$ must be equal to, not less than, $180$ .

Choice D is incorrect. $x + y$ must be equal to, not greater than, $180$ .