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

Intersecting lines r, s, and t are shown below.

The figure presents three intersecting lines, labeled r, s, and t. Line s begins near the top of the figure and extends downward and slightly to the left. Line t begins to the left of line s, and extends downward and to the right, intersecting line s. Line r begins below line t and to the left of line s. Line r extends upward and to the right, first intersecting line s and then line t. The following angle measures are indicated: the angle formed to the right of line s and above line t measures 106 degrees, and the angle formed to the left of line s and above line r measures x degrees. The angle formed below line t and above line r measures 23 degrees.

What is the value of x ?

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Explanation

The correct answer is 97. The intersecting lines form a triangle, and the angle with measure of $x degrees$ is an exterior angle of this triangle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles of the triangle. One of these angles has measure of $23 degrees$ and the other, which is supplementary to the angle with measure $106 degrees$ , has measure of $180 degrees minus 106 degrees, equals 74 degrees$ . Therefore, the value of x is $23 plus 74, equals 97$ .