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Geometry and Trigonometry / Lines, angles, and triangles Difficulty: Hard

Triangles $P Q R$ and $L M N$ are graphed in the xy-plane. Triangle $P Q R$ has vertices $P$ , $Q$ , and $R$ at $left parenthesis 4 comma 5 right parenthesis$ , $left parenthesis 4 comma 7 right parenthesis$ , and $left parenthesis 6 comma 5 right parenthesis$ , respectively. Triangle $L M N$ has vertices $L$ , $M$ , and $N$ at $left parenthesis 4 comma 5 right parenthesis$ , $left parenthesis 4 comma 7 plus k right parenthesis$ , and $left parenthesis 6 plus k comma 5 right parenthesis$ , respectively, where $k$ is a positive constant. If the measure of $∠ Q$ is $t °$ , what is the measure of $∠ N$ ?

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Explanation

Choice C is correct. Since $upper P equals left parenthesis 4 comma 5 right parenthesis$ and $upper Q equals left parenthesis 4 comma 7 right parenthesis$ , side $P Q$ is parallel to the y-axis and has a length of $2$ . Since $upper P equals left parenthesis 4 comma 5 right parenthesis$ and $upper R equals left parenthesis 6 comma 5 right parenthesis$ , side $P R$ is parallel to the x-axis and has a length of $2$ . Therefore, triangle $P Q R$ is a right isosceles triangle, where $∠ P$ has measure $90 degree$ and $∠ Q$ and $∠ R$ each have measure $45 degree$ . It follows that if the measure of $∠ Q$ is $t °$ , then $t equals 45$ . Since $upper L equals left parenthesis 4 comma 5 right parenthesis$ and $M = (4, 7 + k)$ , side $L M$ is parallel to the y-axis and has a length of $k plus 2$ . Since $upper L equals left parenthesis 4 comma 5 right parenthesis$ and $N = (6 + k, 5)$ , side $L N$ is parallel to the x-axis and has a length of $k plus 2$ . Therefore, triangle $L M N$ is a right isosceles triangle, where $∠ L$ has measure $90 degree$ and $∠ M$ and $∠ N$ each have measure $45 degree$ . Of the given choices, only $left parenthesis 90 minus t right parenthesis degree$ is equal to $45 degree$ , so the measure of $∠ N$ is $left parenthesis 90 minus t right parenthesis degree$ .

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

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

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