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Geometry and Trigonometry Difficulty: Hard

In the figure, $R T = T U$ , the measure of angle $V S T$ is $29 degree$ , and the measure of angle $R V S$ is $41 degree$ . What is the value of $x$ ?

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Explanation

The correct answer is $156$ . In the figure shown, the sum of the measures of angle $U V S$ and angle $R V S$ is $180 degree$ . It’s given that the measure of angle $R V S$ is $41 degree$ . Therefore, the measure of angle $U V S$ is $left parenthesis 180 minus 41 right parenthesis degree$ , or $139 degree$ . The sum of the measures of the interior angles of a triangle is $180 degree$ . In triangle $U V S$ , the measure of angle $U V S$ is $139 degree$ and it's given that the measure of angle $V S T$ is $29 degree$ . Thus, the measure of angle $V U S$ is $(180 - 139 - 29) °$ , or $12 degree$ . It’s given that $R T = T U$ . Therefore, triangle $T U R$ is an isosceles triangle and the measure of $V U S$ is equal to the measure of angle $T R U$ . In triangle $T U R$ , the measure of angle $V U S$ is $12 degree$ and the measure of angle $T R U$ is $12 degree$ . Thus, the measure of angle $U T R$ is $(180 - 12 - 12) °$ , or $156 degree$ . The figure shows that the measure of angle $U T R$ is $x °$ , so the value of $x$ is $156$ .