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Geometry and Trigonometry Difficulty: Easy

The figure presents triangle A B C, where side A C is horizontal, vertex A is to the left of vertex C, vertex B is above side A C, and the measure of angle A B C is labeled x degrees. Side A C extends horizontally to the right to point D, and the angle, B C D, that is above line segment C D and to the right of side B C has a measure of one hundred ten degrees.

In the given figure, $side A C$ extends to point D. If the measure of $angle B A C$ is equal to the measure of $angle B C A$ , what is the value of x ?

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Explanation

Choice D is correct. Since $angle B C D$ and $angle B C A$ form a linear pair of angles, their measures sum to 180°. It’s given that the measure of $angle B C D$ is 110°. Therefore, $110 degrees plus angle B C A, equals 180 degrees$ . Subtracting 110° from both sides of this equation gives the measure of $angle B C A$ as 70°. It’s also given that the measure of $angle B A C$ is equal to the measure of $angle B C A$ . Thus, the measure of $angle B A C$ is also 70°. The measures of the interior angles of a triangle sum to 180°. Thus, $70 degrees, plus 70 degrees, plus x degrees, equals 180 degrees$ . Combining like terms on the left-hand side of this equation yields $140 degrees plus x degrees, equals 180 degrees$ . Subtracting 140° from both sides of this equation yields $x degrees equals 40 degrees$ , or $x equals 40$ .

Choice A is incorrect. This is the value of the measure of $angle B C D$ . Choice B is incorrect. This is the value of the measure of each of the other two interior angles, $angle B C A$ and $angle B A C$ . Choice C is incorrect and may result from an error made when identifying the relationship between the exterior angle of a triangle and the interior angles of the triangle.