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Geometry and Trigonometry Difficulty: Hard

In the figure shown, points $Q$ , $R$ , $S$ , and $T$ lie on line segment $P V$ , and line segment $R U$ intersects line segment $S X$ at point $W$ . The measure of $∠ S Q X$ is $48 degree$ , the measure of $∠ S X Q$ is $86 degree$ , the measure of $∠ S W U$ is $85 degree$ , and the measure of $∠ V T U$ is $162 degree$ . What is the measure, in degrees, of $∠ T U R$ ?

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Explanation

The correct answer is $123$ . The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle is $180$ degrees. It's given that the measure of $∠ S Q X$ is $48 degree$ and the measure of $∠ S X Q$ is $86 degree$ . Since points $S$ , $Q$ , and $X$ form a triangle, it follows from the triangle angle sum theorem that the measure, in degrees, of $∠ Q S X$ is $180 - 48 - 86$ , or $46$ . It's also given that the measure of $∠ S W U$ is $85 degree$ . Since $∠ S W U$ and $∠ S W R$ are supplementary angles, the sum of their measures is $180$ degrees. It follows that the measure, in degrees, of $∠ S W R$ is $180 minus 85$ , or $95$ . Since points $R$ , $S$ , and $W$ form a triangle, and $∠ R S W$ is the same angle as $∠ Q S X$ , it follows from the triangle angle sum theorem that the measure, in degrees, of $∠ W R S$ is $180 - 46 - 95$ , or $39$ . It's given that the measure of $∠ V T U$ is $162 degree$ . Since $∠ V T U$ and $∠ S T U$ are supplementary angles, the sum of their measures is $180$ degrees. It follows that the measure, in degrees, of $∠ S T U$ is $180 minus 162$ , or $18$ . Since points $R$ , $T$ , and $U$ form a triangle, and $∠ U R T$ is the same angle as $∠ W R S$ , it follows from the triangle angle sum theorem that the measure, in degrees, of $∠ T U R$ is $180 - 39 - 18$ , or $123$ .