Can you help me with Always, Sometimes, Never questions??
I have answered all of these, but I am not sure if they are right.
1. The base angle of an isosceles triangle are acute----Always
2. When the altitude and the median are drawn from the same vertex of a triangle, the altitude is longer than the median-----Always
3. In triangle ABC, if AB is greater than AC, then <C is greater than <B-----Always
4. All three altitudes of a triangle lie outside the triangle----Sometimes
5. If an exterior angle of a triangle is obtuse, then the triangle is acute---Sometimes
You should rethink 4.
2. is 'sometimes'. Consider the isosceles case.
3. is true because of the law of sines
Sure, I can help you with Always, Sometimes, Never questions. Let's go through each of the statements one by one and determine if the answers are correct.
1. The base angles of an isosceles triangle are acute.
To determine if this statement is true, we need to recall the properties of an isosceles triangle. An isosceles triangle has two sides of equal length and two equal angles. The base angles are the angles opposite the equal sides. Since the sum of the angles in a triangle is always 180 degrees, and if the triangle has two equal angles, those angles must be less than 90 degrees, which means they are acute. Therefore, the statement "The base angles of an isosceles triangle are acute" is always true.
2. When the altitude and the median are drawn from the same vertex of a triangle, the altitude is longer than the median.
To determine if this statement is true, we need to understand the definitions of altitude and median in a triangle. An altitude is a line segment drawn from a vertex of a triangle perpendicular to the opposite side. A median is a line segment drawn from a vertex to the midpoint of the opposite side.
The statement is saying that when the altitude and median are drawn from the same vertex, the altitude is always longer than the median. However, this statement is not always true. There are cases in which the altitude and median can be equal in length, such as in an equilateral triangle. Therefore, the statement "When the altitude and the median are drawn from the same vertex of a triangle, the altitude is longer than the median" is not always true.
3. In triangle ABC, if AB is greater than AC, then angle C is greater than angle B.
To determine if this statement is true, we need to consider the relationship between the lengths of the sides and the corresponding angles in a triangle. In a triangle, the side opposite the larger angle is always longer than the side opposite the smaller angle. Therefore, if AB is greater than AC, then angle C must be greater than angle B. This statement is always true.
4. All three altitudes of a triangle lie outside the triangle.
To determine if this statement is true, we need to understand the definition of an altitude. An altitude is a line segment drawn from a vertex of a triangle perpendicular to the opposite side. In some cases, all three altitudes of a triangle may lie outside the triangle, such as in an obtuse-angled triangle. However, in other cases, at least one altitude can intersect with the triangle, such as in an acute or right-angled triangle. Therefore, the statement "All three altitudes of a triangle lie outside the triangle" is not always true.
5. If an exterior angle of a triangle is obtuse, then the triangle is acute.
To determine if this statement is true, we need to understand the relationship between the exterior angles and the interior angles of a triangle. The sum of the measures of the exterior angles of a triangle is always 360 degrees. Since the exterior angle is obtuse (greater than 90 degrees), the sum of the two interior angles adjacent to it must be less than 180 degrees (as the sum of an interior and exterior angle is always 180 degrees). Therefore, the other two interior angles must be acute. Hence, if an exterior angle of a triangle is obtuse, then the triangle is always acute. This statement is always true.
I hope this explanation helps clarify the answers to the Always, Sometimes, Never questions you provided. If you have any further questions, feel free to ask!