From left to right, horizontal line segment upper P upper V includes the following points: Upper P Upper Q Upper

Question

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 $∠ T U R$ ?

Accepted Answer

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 &#8736; S Q X is 48 &#176; and the measure of &#8736; S X Q is 86 &#176; . Since points S , Q , and X form a triangle, it follows from the triangle angle sum theorem that the measure, in degrees, of &#8736; Q S X is 180 - 48 - 86 , or 46 . It's also given that the measure of &#8736; S W U is 85 &#176; . Since &#8736; S W U and &#8736; S W R are supplementary angles, the sum of their measures is 180 degrees. It follows that the measure, in degrees, of &#8736; S W R is 180 - 85 , or 95 . Since points R , S , and W form a triangle, and &#8736; R S W is the same angle as &#8736; Q S X , it follows from the triangle angle sum theorem that the measure, in degrees, of &#8736; W R S is 180 - 46 - 95 , or 39 . It's given that the measure of &#8736; V T U is 162 &#176; . Since &#8736; V T U and &#8736; S T U are supplementary angles, the sum of their measures is 180 degrees. It follows that the measure, in degrees, of &#8736; S T U is 180 - 162 , or 18 . Since points R , T , and U form a triangle, and &#8736; U R T is the same angle as &#8736; W R S , it follows from the triangle angle sum theorem that the measure, in degrees, of &#8736; T U R is 180 - 39 - 18 , or 123 .

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