Final answer:
Triangles can be constructed under conditions a, c, and d, as they meet the criteria for the sum of angles and lengths of sides. Conditions b and e do not meet these criteria, and thus, triangles cannot be constructed with those specifications.
Step-by-step explanation:
To determine whether it is possible to construct a triangle under certain conditions, we need to consider the properties of triangles. For a triangle to exist, the sum of its angles must equal 180 degrees, and in the case of side lengths, the sum of the lengths of any two sides must be greater than the length of the third side.
- a. A triangle with angle measures 40°, 60°, and 80°: This set of angles adds up to 180° (40+60+80=180), so a triangle can indeed be constructed.
- b. A triangle with side lengths 3 cm, 7 cm, and 11 cm: To see if these sides can form a triangle, we check if the sum of the lengths of any two sides is greater than the length of the third side. Here, 3+7=10, which is not greater than 11, so a triangle cannot be constructed with these side lengths.
- c. A triangle with side lengths 6 in, 13 in, and 12 in: Checking the side lengths, we find 6+12=18, which is greater than 13. Similarly, 6+13=19 and 13+12=25 are both greater than the remaining side. Therefore, a triangle can be constructed.
- d. A triangle with side lengths 8 cm and 5 cm and a 90° angle: This describes a right triangle where we have two sides and a right angle. We can construct a triangle by determining the hypotenuse using the Pythagorean theorem (a² + b² = c²), which yields c=√(8² + 5²) = √(64 + 25) = √89. Therefore, a triangle can be constructed.
- e. A triangle with angle measures 70° and 120°, and a 9 cm side length: The sum of the given angles is 70+120=190°, which exceeds 180°. Thus, a triangle with these angle measures cannot be constructed.