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

In triangle $P Q R$ , $Q R$ is extended to point $S$ . The measure of $∠ P Q R$ is $132 degree$ , and the measure of $∠ P R S$ is $163 degree$ . What is the measure of $∠ Q P R$ ?

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Explanation

Choice B is correct. In the figure shown, since $\bar{Q S}$ is a line segment, the sum of the measures of $∠ P R S$ and $∠ P R Q$ is $180 degree$ . It's given that the measure of $∠ P R S$ is $163 degree$ . Thus, the measure of $∠ P R Q$ is $left parenthesis 180 minus 163 right parenthesis degree$ , or $17 degree$ . The sum of the measures of the interior angles of a triangle is $180 degree$ . It's given that the measure of $∠ P Q R$ is $132 degree$ . Therefore, the measure of $∠ Q P R$ is $(180 - 17 - 132) °$ , or $31 degree$ .

Choice A is incorrect. This is the measure of the supplement of $∠ P Q R$ , not the measure of $∠ Q P R$ .

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

Choice D is incorrect. This is the measure of $∠ P R Q$ , not the measure of $∠ Q P R$ .