Geometry ---- Classifying Triangles 10/17/09

Explanation
The given triangle has sides of lengths 3cm, 4cm, and 6cm. A scalene triangle is a triangle in which all three sides have different lengths. Since the sides of this triangle have different lengths, it can be classified as a scalene triangle.

Explanation
The given angle measures of 40°, 60°, and 80° are all less than 90°, which classifies them as acute angles. Acute angles are angles that are less than 90°. Therefore, the correct answer is "acute".

Explanation
The given angle measures are 25°, 10°, and 145°. An obtuse angle is any angle that measures greater than 90° but less than 180°. Since the measure of the largest angle (145°) is greater than 90°, it is an obtuse angle. Therefore, the answer is obtuse.

Explanation
The given side measures of 6in, 5in, and 6in indicate that two sides have the same length, making it an isosceles triangle. An isosceles triangle has at least two sides of equal length.

Explanation
The given angle measures of 30°, 60°, and 90° indicate that one of the angles is a right angle, which measures 90°. A right angle is formed when two lines intersect and create a 90° angle, forming an "L" shape. Therefore, the correct answer is "right."

Explanation
A triangle can never have both a right angle and an obtuse angle at the same time. In a right-angled triangle, one angle measures exactly 90 degrees, while in an obtuse-angled triangle, one angle measures more than 90 degrees. Since these two angles are mutually exclusive, it is not possible for a triangle to simultaneously have both a right angle and an obtuse angle. Therefore, the correct answer is "never."

Explanation
In a right triangle, one of the angles is always 90 degrees. The sum of the measures of the other two angles must be 90 degrees in order for the triangle to be a right triangle. Therefore, the acute angles of a right triangle are always complementary, meaning they add up to 90 degrees.

Explanation
An obtuse triangle is sometimes an isosceles triangle because an obtuse triangle can have two equal sides, which would make it isosceles. However, not all obtuse triangles are isosceles because they can also have three unequal sides. Therefore, the statement "An obtuse triangle is sometimes an isosceles triangle" is correct.

Explanation
An isosceles triangle is a type of triangle that has two sides of equal length, while an equilateral triangle has all three sides of equal length. Since the definition of an isosceles triangle does not include all three sides being equal, it can never be the same as an equilateral triangle, which requires all three sides to be equal. Therefore, the correct answer is "never".

Explanation
The sum of the angles in a triangle is always 180 degrees. This is a fundamental property of triangles and can be proven using various geometric principles. Regardless of the size or shape of the triangle, the sum of its angles will always be 180 degrees.

Explanation
A triangle with two 45° angles is an acute triangle. In an acute triangle, all three angles are less than 90°. Since two angles are already 45°, it means that the third angle must also be less than 90° in order for the sum of all angles to be less than 180°. Thus, the triangle is acute.

Explanation
An obtuse triangle is a triangle that has one angle greater than 90 degrees. In this case, the given triangle has an angle of 102 degrees, which is greater than 90 degrees, making it an obtuse triangle.

Explanation
The correct answer is "aute." A triangle in which all angles measure 60 degrees is known as an equilateral triangle. In an equilateral triangle, all sides are equal in length, and all angles are equal. Therefore, it cannot be obtuse or scalene.

Explanation
The given triangle has side measures of 2m, 4m, and 7m. A scalene triangle is a triangle in which all three sides have different lengths. In this case, since all three sides have different measures, the triangle is scalene.

Explanation
The given triangle has angle measures of 45°, 55°, and 80°. A scalene triangle is a triangle in which all three sides have different lengths. Since the given triangle does not have any equal sides, it can be classified as a scalene triangle.