Question

consider the statement: \in a triangle, the measure of an exterior angle is always greater than the measure of the adjacent interior angle.\ which of the following would serve as a counterexample? \bigcirc acute triangle \bigcirc altitude of a triangle \bigcirc equilateral triangle \bigcirc right triangle

Explanation:

A counterexample disproves a statement. The statement claims an exterior angle of a triangle is always greater than the adjacent interior angle. In a right triangle, the adjacent interior angle to a right - angled exterior angle (if we consider the exterior angle formed by extending a leg of the right angle) is the right angle itself (90 degrees). The exterior angle here would also be 90 degrees (since a linear pair with a 90 - degree angle is 90 degrees), so it is not greater. An acute triangle has all interior angles less than 90, so exterior angles (supplementary to acute angles) are obtuse and greater. An equilateral triangle has 60 - degree interior angles, and exterior angles are 120 degrees, which are greater. The "altitude of a triangle" is not a type of triangle, so it can't be a counterexample. Wait, no - let's re - evaluate. Wait, the adjacent interior angle and the exterior angle are supplementary (sum to 180 degrees). Wait, no, the exterior angle is equal to the sum of the two non - adjacent interior angles. But the statement is about being greater than the adjacent interior angle. The adjacent interior angle and the exterior angle are supplementary, so if the adjacent interior angle is, say, 90 degrees (in a right triangle), the exterior angle is also 90 degrees (since 180 - 90=90). So in a right triangle, the exterior angle (adjacent to the right angle) is equal to the adjacent interior angle, not greater. So a right triangle serves as a counterexample. The other options: acute triangle - interior angles are less than 90, so exterior angles (180 - interior) are greater than 90, so greater than adjacent interior. Equilateral triangle - interior 60, exterior 120, greater. Altitude is not a triangle type. So the right triangle is the counterexample.

Answer:

D. Right triangle (assuming the options are labeled as A. Acute triangle, B. Altitude of a triangle, C. Equilateral triangle, D. Right triangle)