Is it always true, sometimes true, or never true that the centroid is outside its associated triangle?

Is it always true, sometimes true, or never true that two complementary angles have a common vertex?

always; sometimes

Is this statement always true, sometimes true, or never true? give angle ABC and DEF, if angle A = angle D, angle B = angle E, and angle C = angle F, then angle ABC is congruent to angle DEF

For the first question, it is never true that the centroid is outside its associated triangle. The centroid of a triangle is always located inside the triangle.

For the second question, it is always true that two complementary angles have a common vertex. Complementary angles are defined as two angles that add up to 90 degrees, and by definition, they share a common vertex.

To determine whether the statement is always true, sometimes true, or never true, we need to analyze the properties of the given concepts.

1. Regarding the centroid: The centroid is the point of intersection of the medians of a triangle. A median is a line segment connecting a vertex of a triangle to the midpoint of the opposing side.

It is always true that the centroid is located inside the triangle. This is because the medians divide each other in the ratio of 2:1, meaning that the centroid is closer to the 3/4 distance along each median from the vertex connected to the median. As a result, it will always fall within the bounds of the triangle.

Therefore, the statement "the centroid is outside its associated triangle" is never true.

2. Regarding complementary angles: Two angles are considered complementary if their sum is equal to 90 degrees.

It is sometimes true that two complementary angles have a common vertex. This occurs when two angles are adjacent to each other, sharing a common vertex and a common side. In this case, the non-common sides of each angle form a 90-degree angle.

However, it is not always true that complementary angles have a common vertex. For example, angles 30 degrees and 60 degrees are complementary angles, but they do not share a common vertex since they are not adjacent angles.

Therefore, the statement "two complementary angles have a common vertex" is sometimes true.