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Geometry and Trigonometry Difficulty: Easy

The figure presents 3 lines intersecting at point P. Six angles are formed at the point of intersection. Of the six angles, one angle is labeled x degrees, and counterclockwise angles are labeled as follows: unlabeled, y degrees, unlabeled, z degrees, and unlabeled. A note states that the figure is not drawn to scale.

In the figure, three lines intersect at point P. If $x equals 65$ and $y equals 75$ , what is the value of z ?

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Explanation

Choice C is correct. The angle that is shown as lying between the y° angle and the z° angle is a vertical angle with the x° angle. Since vertical angles are congruent and $x equals 65$ , the angle between the y° angle and the z° angle measures 65°. Since the 65° angle, the y° angle, and the z° angle are adjacent and form a straight angle, it follows that the sum of the measures of these three angles is 180°, which is represented by the equation $65 degrees, plus y degrees, plus z degrees, equals 180 degrees$ . It’s given that y = 75. Substituting 75 for y yields $65 degrees, plus 75 degrees, plus z degrees, equals 180 degrees$ , which can be rewritten as $140 degrees, plus z degrees, equals 180 degrees$ . Subtracting 140° from both sides of this equation yields $z degrees equals 40 degrees$ . Therefore, $z equals 40$ .

Choice A is incorrect and may result from finding the value of $x plus y$ rather than z. Choices B and D are incorrect and may result from conceptual or computational errors.