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- 1. Multiple Choice
Determine whether the following statement is Always, Sometimes, or Never true.
The smallest angle of a right triangle is across from the hypotenuse.
Never. The hypotenuse is the longest side of a right triangle, so it will be across from the largest angle, which is a right angle.
Sometimes. The hypotenuse is across from the right angle of a right triangle, so it will only be across from the smallest angle when the right angle is the smallest angle of the triangle.
Always. The hypotenuse is the longest side and the longest side of any triangle is across from the smallest angle of the triangle.
- 2. Multiple Choice
Determine whether the following statement is Always, Sometimes, or Never true.
The two acute angles of a right triangle are complementary.
Never. Acute angles each have a measure less than 90o, so two acute angles cannot be complementary.
Sometimes. A triangle with two congruent sides will have two complementary angles, so the acute angles will only be complementary if the right triangle is isosceles.
Always. A right triangle has one right angle and two acute angles. Since the three angles must add up to 180o, the two acute angles must add up to 90o.
- 3. Multiple Choice
Determine whether the following statement is Always, Sometimes, or Never true.
The smallest angle measure in a triangle is across from the longest side length in the triangle.
Never. The longest side of any triangle must be across from the largest angle in the triangle.
Sometimes. In an equilateral triangle, all sides and angles are congruent, so the smallest angle is simultaneously the largest angle.
Always. The longest side of any triangle must be across from the smallest angle in the triangle.
- 4. Multiple Choice
Determine whether the following statement is Always, Sometimes, or Never true.
The vertex angle of an isosceles triangle is the largest angle in the triangle.
Never. If a triangle has two congruent sides, it will also have two congruent angles. If the vertex angle had the largest measure, the three angles would add up to more than 180o.
Sometimes. Isosceles triangles can be acute, obtuse, or right. In a right or obtuse isosceles triangle, the vertex angle will be the largest angle. In an acute isosceles triangle, the vertex angle will either be the largest or smallest angle.
Always. If a triangle has two congruent sides, it will also have two congruent angles. The congruent angles must be acute, so the vertex angle must have the largest measure in order for the angles to add up to 180o.
- 5. Multiple Choice
Determine whether the following statement is Always, Sometimes, or Never true.
The longest side of an obtuse triangle is at least twice the length of the next longest side.
Never. In order for three lengths to form a triangle, the sum of the two shorter side lengths must be greater than the longest side length. This means that one of the two shorter lengths needs to be more than half the length of the longest side.
Sometimes. In an obtuse triangle, the longest side squared must be greater than the sum of the two shorter sides. So 142 > 72 + 62 and 152 > 112 + 102 will both result in obtuse triangles.
Always. In an obtuse triangle, the longest side squared must be greater than the sum of the squares of the two shorter sides, so the longest side needs to be significantly longer than either of the other sides.
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