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

The figure presents triangle R T U, with R U horizontal, and U to the right of R. A point V is on R U, such that V is closer to R than it is to U. Vertex T is above R U. The angle at T is labeled 114 degrees. Side U T is extended to point S, above vertex R, and line segment S V is drawn. The angle to the right of S V and below S T is labeled 31 degrees. The angle to the left of S V and above R V is labeled x degrees.

In the figure above, $R T equals, T U$ . What is the value of x ?

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Explanation

Choice C is correct. Since $the length of side R T equals the length of side T U$ , it follows that $triangle R T U$ is an isosceles triangle with base RU. Therefore, $angle T R U$ and $angle T U R$ are the base angles of an isosceles triangle and are congruent. Let the measures of both $angle T R U$ and $angle T U R$ be $t degrees$ . According to the triangle sum theorem, the sum of the measures of the three angles of a triangle is $180 degrees$ . Therefore, $114 degrees plus 2 t degrees, equals 180 degrees$ , so $t equals 33$ .

Note that $angle T U R$ is the same angle as $angle S U V$ . Thus, the measure of $angle S U V$ is $33 degrees$ . According to the triangle exterior angle theorem, an external angle of a triangle is equal to the sum of the opposite interior angles. Therefore, $x degrees$ is equal to the sum of the measures of $angle V S U$ and $angle S U V$ ; that is, $31 degrees plus 33 degrees, equals 64 degrees$ . Thus, the value of x is 64.

Choice B is incorrect. This is the measure of $angle S T R$ , but $angle S T R$ is not congruent to $angle S V R$ . Choices A and D are incorrect and may result from a calculation error.